calculus 3: summary and formulas
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Calculus 3 Summary and Formulas
One of the fundamental areas covered in Calculus 3 is performing calculus based operations on objects called vectors. Vectors are used to represent quantities that have both a direction and magnitude. They play a very important role in virtually all aspects of science and engineering. Therefore, our first series of lessons were used to develop the basic properties of vectors.
Basic Vector Definitions
A vector is used to express a quantity with both a magnitude and direction. A vector, đ, is determined by a basepoint đ and a terminal point đ as follows
đ = đđââââ â = âš (đđ„ â đđ„), ( đđŠ â đđŠ), ( đđ§ â đđ§) â© = âš đŁđ„, đŁđŠ , đŁđ§â©
Where, đŁđ„, đŁđŠ are called the components of the vector.
The magnitude, i.e. length, of a vector, đ, is referred to as the norm and is given by
âđâ = âđŁđ„2 + đŁđŠ
2
A unit-vector is a vector that has a magnitude of one and can be expressed as follows:
ïżœÌïżœ = âšcos đ , sin(đ)â© Furthermore, any vector, đ, can be scaled to be a unit vector as follows:
ïżœÌïżœ =1
âđâđ
Vector Operations Using Components
If đ = âšđđ„, đđŠâ© and b= âšđđ„ , đđŠâ© then:
i. Addition đ + đ = âšđđ„ + đđ„, đđŠ + đđŠâ©
ii. Subtraction đ â đ = âšđđ„ â đđ„, đđŠ â đđŠâ©
iii. Scalar Multiplication đđ = âšđđđ„, đđđŠ â©
iv. Addition Identity đ + đ = đ + đ = đ
**Note: The operations are shown in 2 dimensions but apply equally in 3 dimensions. Basic Properties of Vector Algebra
For all vectors đ, đ, đ and for all scalars, đ
i. Commutative Law đ + đ = đ + đ
ii. Associative Law đ + (đ + đ) = (đ + đ) + đ
iii. Distributive law for Scalars đ(đ + đ) = đđ + đđ
Standard Basis Vectors for Rectangular Coordinate System in 3-Dimensions
đ = âš1, 0, 0â©
đ = âš0, 1, 0â©
đ = âš0, 0, 1â© y
x
z
i
jk
All vectors in can be written as a linear combination of the basic vectors.
đ = âšđ, đ, đâ© = đđ + đđ + đđ
Triangle Inequality
For any two vectors đ and đ.
âđ + đâ †âđâ + âđâ The equality holds when đ = đ or đ = đ, or if đ = đđ, where đ > 0.
The Dot Product
Given two vectors, đ and đ, as well as the angle, đ, between the two vectors. The dot product can be equivalently be defined in the following two ways:
đ â đ = âđââđâ đđđ (đ)
đ â đ = (đđ„đđ„ + đđŠđđŠ + đđ§đđ§)
Furthermore, if the angle is unknown it may be found as follows:
đ = cosâ1 (đ â đ
âđââđâ) = cosâ1 (
đđ„đđ„ + đđŠđđŠ + đđ§đđ§
âđââđâ)
The angle between two vectors is chosen to satisfy 0 †đ †đ
Properties of the Dot Product
1. Commutative Property: đ â đ = đ â đ
2. Zero Property: đ â 0 = đ
3. Scalar Multiplication Property: đ(đ â đ) = (đđ) â đ = đ â (đđ)
4. Distributive Property: đ â (đ + đ) = đ â đ + đ â đ
5. Relation to Length: đ â đ = âđâđ
Geometric Properties of The Dot Product
đ â đ > đ
a
bΞ
The angle between the two
vectors is acute, i.e. 0° †đ <
90°
đ â đ < đ
a
b
Ξ
The angle between the two
vectors is obtuse, i.e. 90° <
đ †180°
đ â đ = đ
a
b
The angle between the two
vectors is 90°. Note: We use the word orthogonal to refer to vectors
that form a đđ° angle.
Projection Vector
The projection of a vector đ onto the vector đ is the vector, đ||đ given by
đ||đ = (đ â đ
đ â đ)đ = (
đ â đ
âđâ2)đ = (
đ â đ
âđâ) ïżœÌïżœ
The scalar, (đâđ
âđâ) = âđâ đđđ (đ), is called the component of đ along đ.
ba
a||b
Vector Decomposition
Any vector, đ can be decomposed into two orthogonal component vectors with respect to another vector, đ as:
đ = đâ„đ + đâ„đ Where the parallel projection is given above, and the perpendicular projection is found as:
đâ„đ = đ â đâ„đ
The Cross Product
The cross product of two vectors, đ = âšđđ„, đđŠ, đđ§â© and đ = âšđđ„, đđŠ, đđ§â© is a new vector đ,
given as
đ = đ Ă đ = |
ïżœÌïżœ đÌ ïżœÌïżœđđ„ đđŠ đđ§
đđ„ đđŠ đđ§
| = ïżœÌïżœ |đđŠ đđ§
đđŠ đđ§| â đÌ |
đđ„ đđ§
đđ„ đđ§| + ïżœÌïżœ |
đđ„ đđŠ
đđ„ đđŠ|
= (đđŠđđ§ â đđŠđđ§)ïżœÌïżœ â (đđ„đđ§ â đđ„đđ§)ïżœÌïżœ + (đđ„đđŠ â đđ„đđŠ)ïżœÌïżœ
Geometric Interpretation of the Cross Product
Given two vectors, đ and đ, the cross product, đ Ă đ is a unique vector with the following properties.
i. đ Ă đ is orthogonal to đ and đ.
ii. The length of đ Ă đ is âđââđâ sin(đ), where đ is the angle between đ and đ and is chosen to satisfy 0 †đ †đ.
The Right-Hand Rule
The right-hand rule can be stated as: The vector, đ Ă đ, is orthogonal to a plane that is parallel to đ and đ. Furthermore, when the fingers of your right hand curl from đ to đ, your thumb points to the side of the plane for which the resulting vector points.
Properties of the Cross Product
i. đ Ă đ = âđ Ă đ
ii. đ Ă đ = đ
iii. đ Ă đ = đ is and only if đ = đđ for some scalar đ or đ = đ
iv. đ(đ Ă đ) = (đđ) Ă đ = đ Ă (đđ)
v. đ Ă (đ + đ) = (đ Ă đ) + (đ Ă đ), (đ + đ) Ă đ = (đ Ă đ) + (đ Ă đ)
Properties of the Cross Product
vi. đ Ă đ = âđ Ă đ
vii. đ Ă đ = đ
viii. đ Ă đ = đ is and only if đ = đđ for some scalar đ or đ = đ
ix. đ(đ Ă đ) = (đđ) Ă đ = đ Ă (đđ)
x. đ Ă (đ + đ) = (đ Ă đ) + (đ Ă đ), (đ + đ) Ă đ = (đ Ă đ) + (đ Ă đ)
Cross Product of The Standard Basis Vectors
ïżœÌïżœ Ă đÌ = ïżœÌïżœ đÌ Ă ïżœÌïżœ = ïżœÌïżœ ïżœÌïżœ Ă ïżœÌïżœ = đÌ
đÌ Ă ïżœÌïżœ = âïżœÌïżœ ïżœÌïżœ Ă đÌ = âïżœÌïżœ ïżœÌïżœ Ă ïżœÌïżœ = âđÌ
ïżœÌïżœ Ă ïżœÌïżœ = đ đÌ Ă đÌ = đ ïżœÌïżœ Ă ïżœÌïżœ = đ
Area and Cross Product
If đ« is the parallelogram formed by the vectors đ and đ, then the area, đŽđ«, can be found as đŽđ« = âđ Ă đâ
Volume and Cross Product
If đ« is the parallelepiped formed by the vectors đ, đ and đ, then the volume, đđ«, can be found as
đđ« = |(đ) â (đ Ă đ)| Where, (đ) â (đ Ă đ) is referred to as the vector triple product and can be represented as
(đ) â (đ Ă đ) = |
đđ„ đđŠ đđ§
đđ„ đđŠ đđ§
đđ„ đđŠ đđ§
| = đ đđ (đđđ)
Equation of a Line in đčđ (Point-Direction Form)
The line â through the point (đ„0, đŠ0, đ§0) in the directions of đ = âšđ, đ, đâ© can be described in the following ways: Vector Parameterization:
đ(đĄ) = đđ + đĄđ = âšđ„0, đŠ0, đ§0â© + đĄâšđ, đ, đâ© Parametric Equations:
đ„(đĄ) = đ„0 + đđĄ đŠ(đĄ) = đŠ0 + đđĄ đ§(đĄ) = đ§ + đđĄ
Where (ââ < đĄ < â)
Parallel, Perpendicular, and Intersecting Lines
Two lines are parallel when the cross product of their direction vectors is zero. đđ Ă đđ = đ
Two lines are perpendicular when the dot product of their direction vectors is zero.
đđ â đđ = 0 The point of intersection between two lines can be found by setting the equations of the lines equal to one another for different values of the parameter đĄ.
đđ(đĄ1) = đđ(đĄ2) If an intersection point does not exist and the lines are not parallel, we refer to them as skewed.
Equation of a Plane in đčđ (Point-Normal Form)
The plane đ« through the point (đ„0, đŠ0, đ§0) with a normal vector đ = âšđ, đ, đâ© can be described in the following ways: Vector Form:
đ â âšđ„, đŠ, đ§â© = đ Scalar Form:
đ(đ„ â đ„0) + đ(đŠ â đŠ0) + đ( đ§ â đ§0) = 0
đđ„ + đđŠ + đđ§ = đ
Where, đ = đđ„0 + đđŠ0 + đđ§0
Parallel and Intersecting Planes
Two planes are parallel when the cross product of their normal vectors is zero. đđ Ă đđ = đ
Two planes are perpendicular when the dot product of their normal vectors is zero.
đđ â đđ = 0 When two planes are not parallel, they intersect along a line, Line of Intersection (LOS). The direction vector of the LOS is given as
đ = đđ Ă đđ
Parallel and Perpendicular Lines and Planes
A plane and a line are parallel when the dot product of the normal and direction vector is zero.
đ â đ = 0 A plane and a line are perpendicular when the cross product of the normal and direction vector is zero.
đ Ă đ = đ
Calculus 1 and 2 focused on single variable calculus. As you can see from the vector summary above, Calculus 3 studies multivariable functions also. Quadric surfaces are quadratic equations in three variables and are common in Calculus 3. In addition, other coordinates system are utilized in Calculus 3, usually to render a particular problem much easier
Quadric Surface
A quadric surface is defined by a quadratic equation in three variables. The general form is đŽđ„2 + đ”đŠ2 + đ¶đ§2 + đ·đ„đŠ + đžđŠđ§ + đčđ§đ„ + đđ„ + đđŠ + đđ§ + đ = 0
When đ· = đž = đč = đ = đ = đ = 0, the quadric axes are parallel to the coordinate axes and the surface is centered at (0, 0). When this is the case the equations are said to be in standard form.
Quadric Surfaces in Standard Form
1. Sphere: Centered at (0, 0, 0) with a radius đ.
đ„2 + đŠ2 + đ§2 = đ2
2. Ellipsoid: Centered at (0, 0, 0) with đ„, đŠ, and đ§ âradiusâ equal to đ, đ, and đ respectively.
(đ„
đ)2
+ (đŠ
đ)2
+ (đ§
đ)2
= 1
3. Hyperboloid:
One Sheet (đ„
đ)2
+ (đŠ
đ)2
= (đ§
đ)2
+ 1
Two Sheets (đ„
đ)2
+ (đŠ
đ)2
= (đ§
đ)2
â 1
Elliptical Cone (limiting case of one sheet) (đ„
đ)2
+ (đŠ
đ)2
= (đ§
đ)2
+ 0
4. Paraboloid:
Elliptical (bowl) đ§ = (đ„
đ)2
+ (đŠ
đ)2
Hyperbolic (Saddle) đ§ = (đ„
đ)2
â (đŠ
đ)2
Trace
A trace is the intersection of a surface with a given plane. A trace can be obtained by âfreezingâ one of the three variables and sketching the resulting 2D equation. Traces can be used to help us to draw the graph of a surface. Horizontal Trace:
Setting đ§ = đ§0 results in a curve in an đ„đŠ oriented plane. Vertical Trace:
Setting đ„ = đ„0 results in a curve in a đŠđ§ oriented plane. Setting đŠ = đŠ
0 results in a curve in an đ„đ§ oriented plane.
Cylindrical Coordinate System
y
x
z
Ξ r
z
P=(x,y,z)
Q=(x,y,0)
A point in cylindrical coordinates is described by: đ = (đ, đ, đ§)
âą đ: The horizontal distance from the origin.
âą đ: The polar angle measured from the positive đ„-axis.
âą đ§: The vertical distance from the origin.
Rectangular and Cylindrical Coordinate Conversion Formulas
Cylindrical to Rectangular Rectangular to Cylindrical
đ„ = đ đđđ (đ)
đŠ = đ đ đđ(đ)
đ§ = đ§
đ = âđ„2 + đŠ2
đ = đĄđđâ1 (đŠ
đ„)
đ§ = đ§
Spherical Coordinate System
x
P
Ξ
Ï
Ï
y
z
y
x
z
Ξ
O
A point in spherical coordinates is described by: đ = (đ, đ, đ)
âą đ: The distance from the origin to the point, đ, where đ â„ 0
âą đ: The angle of the projection for 0Pââ ââ onto
the đ„-đŠ plane, where â180° †đ †180°
âą đ: The angle of declination, which
measures how much the vector, 0Pââ ââ ,
declines form the vertical, where 0° â€
đ †180°
Rectangular and Spherical Coordinate Conversion Formulas
Spherical to Rectangular Rectangular to Spherical
đ„ = đ đ đđ(đ) đđđ (đ)
đŠ = đ đ đđ(đ) đ đđ(đ)
đ§ = đ đđđ (đ)
đ = âđ„2 + đŠ2 + đ§2
đ = đĄđđâ1 (đŠ
đ„)
đ = đđđ â1 (đ§
đ)
Level Surfaces
Level Surfaces are surfaces obtained by setting one of the coordinates to a constant. Rectangular Coordinate System:
âą đ„ = đ¶: Vertically aligned plane parallel to the đŠ-đ§ plane.
âą đŠ = đ¶: Vertically aligned plane parallel to the đ„-đ§ plane.
âą đ§ = đ¶: Horizontally aligned plane parallel to the đ„-đŠ plane Cylindrical Coordinate System:
âą đ = đ¶: Cylinder with radius, đ¶.
âą đ = đ¶: Vertical half plane oriented at an angle, đ¶.
âą đ§ = đ¶: Horizontally aligned plane parallel to the đ„-đŠ plane Spherical Coordinate System:
âą đ = đ¶: Sphere with radius, đ¶
âą đ = đ¶: Vertical half plane oriented at an angle, đ¶.
âą đ = đ¶: Right circular cone with an opening at an angle, đ¶.
Once we understood the basics of vectors our next series of lessons focused on performing calculus on vectors. One of the main applications of vector calculus is the ability to study motion in 3 dimensions.
Vector-Valued Function
A vector-valued function is any function whose domain is a set of real number and whose range is a set of vectors. The variable đĄ is called a parameter, which doesnât necessarily have to represent time, and the functions đ„(đĄ), đŠ(đĄ) and đ§(đĄ) are called the components or coordinate functions.
đ(đĄ) = âšđ„(đĄ), đŠ(đĄ), đ§(đĄ)â© We can also represent the vector parameterization of a path as a curve with a set of parametric equations as
đ(đĄ) = (đ„(đĄ), đŠ(đĄ), đ§(đĄ))
Note: The curve is the set of all points, đ„(đĄ), đŠ(đĄ), đ§(đĄ), as đĄ varies over its domain. However, the path referred to by đ(đĄ) represents the particular way the curve is traversed, e.g. it may traverse the curve several times, reverse direction, move back and forth, etc.
y
x
z
r(t1)
r(t2) r(t3)
r(t) = <x(t), y(t), z(t)>
Projections
Projections of đ(đĄ) onto a plane can help us sketch the underlying curve. We project onto each plane by setting the third coordinate to zero. Projection onto đ„-đŠ plane: Let đ§(đĄ) = 0, đ(đĄ) = âšđ„(đĄ), đŠ(đĄ), 0â©
Projection onto đ„-đ§ plane: Let đŠ(đĄ) = 0, đ(đĄ) = âšđ„(đĄ), 0, đ§(đĄ)â©
Projection onto đŠ-đ§ plane: Let đ„(đĄ) = 0, đ(đĄ) = âš0, đŠ(đĄ), đ§(đĄ)â©
Derivative of Vector-Valued Function
The derivative of the vector-valued function đ(đĄ) = âšđ„(đĄ), đŠ(đĄ), đ§(đĄ)â©, is computed component-wise as
đ
đđĄ(đ(đĄ)) = đâČ(đĄ) = âšđ„âČ(đĄ), đŠâČ(đĄ), đ§âČ(đĄ)â©
Provided each component is differentiable.
Differentiation Rules for Vector-Valued Functions
Sum Rule: đ
đđĄ(đđ(đĄ) + đđ(đĄ)) =
đ
đđĄ(đđ(đĄ)) +
đ
đđĄ(đđ(đĄ))
Constant Multiple Rule: đ
đđĄ(đđ(đĄ)) = đ
đ
đđĄ(đ(đĄ))
Scaler Product Rule: đ
đđĄ(đ(đĄ)đ(đĄ)) = đâČ(đĄ)đ(đĄ) + đ(đĄ)đâČ(đĄ)
Dot Product Rule: đ
đđĄ(đđ(đĄ) â đđ(đĄ)) = đđ
âČ(đĄ) â đđ(đĄ) + đđ(đĄ) â đđâČ(đĄ)
Cross Product Rule: đ
đđĄ(đđ(đĄ) Ă đđ(đĄ)) = đđ
âČ(đĄ) Ă đđ(đĄ) + đđ(đĄ) Ă đđâČ(đĄ)
Chain Rule: đ
đđĄ(đ(đ(đĄ))) = đâČ(đ(đĄ))đâČ(đĄ)
Derivative of Vector-Valued Function as a Tangent Vector
The derivative at đĄ0, đâČ(đĄ0), is a vector that is tangent to the path, đ(đĄ), at đĄ0.
The tangent line to the path, đ(đĄ), at đĄ0 can be written as
đł(đĄ) = đ(đĄ0) + đĄđâČ(đĄ0)
Orthogonality of đ and đâČ when đ has a Constant Length
If đ(đĄ) is a differentiable vector-valued function in đ 2 or đ 3, and if âđ(đĄ)â is constant for all đĄ, then đ(đĄ) â đâČ(đĄ) = 0. That is, đ(đĄ) and đâČ(đĄ) are orthogonal vectors for all đĄ.
Indefinite and Definite Integral of a Vector-Valued Function
The Indefinite Integral of a vector-valued function is defined as
â«đ(đĄ)đđĄ = âšâ«đ„(đĄ)đđĄ ,â« đŠ(đĄ)đđĄ ,â« đ§(đĄ)đđĄâ© + đ
The Definite Integral of a vector-valued function is defined as
â« đ(đĄ)đđĄ = âšâ« đ„(đĄ)đđĄđ
đ
, â« đŠ(đĄ)đđĄđ
đ
, â« đ§(đĄ)đđĄđ
đ
â©đ
đ
Arc Length - The length of a Path (Distance Traveled)
Assume đ(đĄ) is differentiable and đâČ(đĄ) is continuous on [đ, đ]. Then the distance, đ , a particle travels along the path, đ(đĄ), for đ †đĄ †đ is equal to
đ = â« âđâČ(đĄ)âđ
đ
đđĄ = â« âđ„âČ(đĄ)2 + đŠâČ(đĄ)2 + đ§âČ(đĄ)2đ
đ
đđĄ
The distance traveled as a function of đĄ can also be written as
đ (đĄ) = â« âđâČ(đą)âđĄ
đ
đđą
Which, we sometimes refer to as the arc length function.
Position, Velocity, Distance, and Speed Relationships
Given the following: đ(đĄ): The velocity of a particle at time đĄ. đŁ(đĄ): The speed of a particle at time đĄ. đ(đĄ): The position of a particle at time đĄ. đ (đĄ): The distance a particle has traveled at time đĄ.
We can write the following relationships:
The velocity is the time derivative of position: đ(đĄ) = đâČ(đĄ)
The speed is the magnitude of velocity: đŁ(đĄ) = âđ(đĄ)â = âđâČ(đĄ)â
The position is the time integral of velocity: đ(đĄ) = â«đ(đĄ) đđĄ + đ(đ)
The distance traveled, arc length, is the time integral of speed:
đ (đĄ) = â« âđâČ(đą)âđĄ
đ
đđą
Arc Length (Unit Speed) Parameterization
The arc length parameterization of a curve is one in which the speed is unity, i.e. âđ(đ )â = 1. This restriction, âđ(đ )â = 1, allows for the creation of a unique parameterization that focusing on the shape of the curve only and not on the particular way in which it is traversed. Starting with any parameterization, đ(đĄ), we proceed as follows: Step 1: Find the arc length function.
đ = đ(đĄ) = â« âđâČ(đą)âđĄ
đ
đđą
Step 2: Compute the following inverse function.
đĄ = đâ1(đ ) Step 3: Create the new unit speed parameterization as follows:
đ(đ ) = đ(đâ1(đ ))
Curvature
Curvature is a positive numerically positive value that measures how a curve bends. It is defined based using the arc length parametrization of a curve as specified below. Let đ(đ ) be an arc length parameterization and đ» = đ»(đ ) be the unit tangent vector. The curvature at đ(đ ) is defined as follows:
đ (đ ) = âđđ»
đđ â
Where, đ» = đ»(đ ) = đâČ(đ )
Note: This assumes đâČ(đĄ) â 0 for all đĄ.
Curvature Defined for Arbitrary Parameterizations
Alternate forms for computing the curvature can be derived without using the arc length parametrization as shown below. If đ(đĄ) is an arbitrary parameterization, the curvature can be computed with either of the two formulas:
đ (đĄ) =1
đŁ(đĄ)âđđ»
đđĄâ đ (đĄ) =
âđâČ(đĄ) Ă đâČâČ(đĄ)â
âđâČ(đĄ)â3
Curvature of a Graph in a Plane
The curvature of the graph of đŠ = đ(đ„) is equal to
đ (đ„) =|đâČâČ(đ„)|
(1 + (đâČ(đ„))2)3 2â
Frenet Frame
A unit vector that is tangent to a space curve, đ(đĄ), for all đĄ is called the unit tangent vector and is given as
đ»(đĄ) = đâČ(đĄ)
âđâČ(đĄ)â
A unit normal vector that is orthogonal to đ»(đĄ) for all đĄ and points in the direction that the curve is turning is called the unit normal vector and is given as
đ”(đĄ) = đ»âČ(đĄ)
âđ»âČ(đĄ)â
A unit vector that is orthogonal to both đ»(đĄ) and đ”(đĄ) is called a unit binormal vector and is give as
đ©(đĄ) = đ»(đĄ) Ă đ”(đĄ) The three vectors, (đ»,đ”,đ©), are mutually orthogonal and of unit length. Together they form an orthonormal set of vectors, which we refer to as the Frenet Frame. The Frenet frame is a function of the underlying curve and changes from point to point along the curve. As such, it is very useful in analyzing motion of objects in space
x
y
z T
N
B
Osculating Circle
The osculating circle to a plane curve, đ(đĄ), at the point đ is the circle that âbest fitsâ the curve at đ. The center of the circle lies in the direction of the normal vector, đ”, to the curve, and the radius of the circle is called the radius of curvature, đ = 1 đ đâ . The equation of the osculating circle to the plane curve, đ(đĄ), at đ = đ(đĄ0) is given as follows:
đđ(đĄ) = đ(đĄ0) + 1 đ đâ (âšđđđ (đĄ) , đ đđ(đĄ) â© + đ”đ)
P
Tr(t0)
Q
RN
x
y
Motion Describing Quantities
đ(đĄ) : Position Vector â Represents the Position of an Object : đ(đĄ) = âšđ„(đĄ), đŠ(đĄ), đ§(đĄ)â©
đ(đĄ) : Velocity Vector â Rate of change of Position : đ(đĄ) = đâČ(đĄ).
đŁ(đĄ) : Speed â Magnitude of Velocity : đŁ(đĄ) = âđ(đĄ)â
đ(đĄ) : Acceleration Vector - Rate of change of Velocity : đ(đĄ) = đâČ(đĄ) = đâČâČ(đĄ)
Acceleration Vector Decomposition
The acceleration vector for an object traveling along a path is given as
đ(đĄ) = đđđ»(đĄ) + đđđ”(đĄ) Where, đđ = đŁâČ(đĄ), and đđ = đ đŁ2(đĄ)
âą The Tangential Component âencodesâ the change in the speed o Since đđ = đŁâČ(đĄ) the tangential component is zero if the speed is constant.
âą The Normal Component âencodesâ the change in direction o Since đđ = đ đŁ2(đĄ) the normal component is zero if đ = 0, which is the case when
the path does not change direction. The decomposition vectors can also be evaluated using the following formulas.
đđđ»(đĄ) = (đ(đĄ) â đ(đĄ)
âđ(đĄ)âđ)đ(đĄ)
đđđ”(đĄ) = đ(đĄ) â đđđ»(đĄ)
= đ(đĄ) â (đ(đĄ) â đ(đĄ)
âđ(đĄ)âđ)đ(đĄ)
Non-Uniform Circular Motion
đŁâČ(đĄ) = đđ = đ(đĄ) â đ»(đĄ) = âđ(đĄ)ââđ»(đĄ)â đđđ (đ) (đš) (đ©) (đȘ)
Ξ
Ξ
Ξ
A. đ = 90°: Therefore, đđđ (đ) = 0 and đŁâČ(đĄ) = 0. The particles speed is constant, which results on uniform circular motion as shown in example 4.
B. đ < 90°: Therefore, đđđ (đ) > 0 and đŁâČ(đĄ) > 0. The particles speed is increasing.
C. 90° < đ < 180°: Therefore, đđđ (đ) < 0 and đŁâČ(đĄ) < 0. The particles speed is decreasing
Next series of lessons focused on differentiation of multivariable scalar functions. One way vectors play a role for multivariable differentiation is the derivative of multivariable functions are directional in the sense that the object has both a magnitude and direction.
Multivariable Functions
A multivariable function is one that takes đ real variables as inputs, (đ„1, đ„2, . . . , đ„đ), and assigns a single value, đŠ, to each đ-tuple (đ„1, đ„2, . . . , đ„đ) in a domain in đ đ. The range is the set of all đŠ values for the (đ„1, đ„2, . . . , đ„đ) in the domain.
âą (đ„1, đ„2, . . . , đ„đ) are called the independent variables.
âą đŠ is the dependent variable.
The function is represented as
đŠ = đ(đ„1, đ„2, . . . , đ„đ)
Traces, Level Curves, and Contour Maps
âą Vertical Trace o The intersection of the graph with a vertical plane obtained by setting đ„ or đŠ to đ.
âȘ Vertical trace parallel with the đŠ-đ§ plane: Consists of all points (đ, đŠ, đ(đ, đŠ) ). âȘ Vertical trace parallel with the đ„-đ§ plane: Consists of all points (đ„, đ, đ(đ„, đ) ).
âą Horizontal Trace o The intersection of the graph with a horizontal plane obtained by setting đ(đ„, đŠ) to đ.
âȘ Horizontal traces are parallel to the đ„-đŠ plane and consist of all points (đ„, đŠ, đ ).
âą Level Curve o The projection of a horizontal trace in the đ„-đŠ plane.
âȘ The curve đ(đ„, đŠ) = đ in the đ„-đŠ plane.
âą Contour Map o A plot in the đ„-đŠ plane showing level curves đ(đ„, đŠ) = đ for equally spaced values of đ.
âą Contour Interval o The interval, đ, between the level curves in a contour map. o When moving from one level curve to the next, the value of đ(đ„, đŠ) changes by ±đ.
Contour Maps and Rate of Change
âą The level curves on a contour map are drawn at equally spaced changes in đ(đ„, đŠ).
âą The spacing between level curves on a contour map indicates the âsteepnessâ of the change in đ(đ„, đŠ).
âą The average rate of change from a point đ to a point đ on a contour map, đŽâđâđ,is
đŽâđâđ=â đčđąđđđĄđąđđđ đŁđđđąđ
â đ»đđđđ§đđđđĄđ đ·đđ đĄđđđđ
When the function represents the physical height of an area, we usually say
đŽâđâđ=â đŽđđĄđđĄđąđđ
â đ»đđđđ§đđđđĄđ đ·đđ đĄđđđđ
Partial Derivatives
Partial derivatives are defined for multivariable functions. They are derivatives with respect to one of the variables. Specifically, when computing a partial derivative for a generic multivariable function, e.g. đ(đ„1, đ„2, . . . , đ„đ), with respect to a specific variable, e.g. đ„1, we treat all other variables, e.g. đ„2, . . . đ„đ, as if they are constant values.
Partial Derivatives for Two Variable Functions
The partial derivative of đ(đ„, đŠ) with respect to đ„ is defined
đđ„(đ„, đŠ) = đđđââ0
{đ(đ„ + â, đŠ) â đ(đ„, đŠ)
â}
Equivalent Notations
đđ„(đ„, đŠ) = đđ„ =đ
đđ„đ(đ„, đŠ) =
đđ
đđ„
The partial derivative of đ(đ„, đŠ) with respect to đŠ is defined
đđŠ(đ„, đŠ) = đđđââ0
{đ(đ„, đŠ + â) â đ(đ„, đŠ)
â}
Equivalent Notations
đđŠ(đ„, đŠ) = đđŠ =đ
đđŠđ(đ„, đŠ) =
đđ
đđŠ
Partial Differentiation Algebraic Rules
Sum Rule: đ
đđ„(đ ± đ) =
đđ
đđ„±
đđ
đđ„
Product Rule: đ
đđ„(đđ) =
đđ
đđ„đ + đ
đđ
đđ„
Quotient Rule: đ
đđ„(đ
đ) =
đđđđ„
đ â đđđđđ„
đ2
Higher Order Partial Derivatives and Clairautâs Theorem
Similar to single variable derivatives, higher order partial derivatives are derivatives of derivatives. For example, the second order partial derivative with respect to đ„ is
đđ„đ„ = đ
đđ„(đđ
đđ„)
For multivariable functions we also have what are called mixed partials, e.g. đđ„đŠ and đđŠđ„.
Clairautâs Theorem states that the order in which we choose to perform the derivatives does not matter, provided the mixed partials are continuous functions. In other words, for two variable functions the theorem guarantees the following:
đđ„đŠ = đđŠđ„
Equation of the Tangent Plane and Normal Line
The tangent plane to the surface, đ(đ„, đŠ), at the point (đ„0, đŠ0, đ§0) is given by
đ§ = đđ„(đ„0, đŠ0)(đ„ â đ„0) + đđŠ(đ„0, đŠ0)( đŠ â đŠ0) + đ§0
The normal line to the surface is given as
đ(đĄ) = âš(đ„0 + đđ„(đ„0, đŠ0)đĄ), (đŠ0 + đđŠ(đ„0, đŠ0)đĄ), (đ§0 â đĄ)â©
Linear Approximation and Differentials
The linear approximation of đ(đ„, đŠ) around the point (đ, đ, đ(đ, đ)) is given by the equation
of the tangent plane at that point.
đż(đ„, đŠ) = đđ„(đ, đ)(đ„ â đ) + đđŠ(đ, đ)( đŠ â đ) + đ(đ, đ)
The value of a function, đ(đ„, đŠ), at (đ + âđ„, đ + âđŠ) can be approximated by this linearization, đż(đ„, đŠ), as
đ(đ + âđ„, đ + âđŠ) â đđ„(đ, đ)âđ„ + đđŠ(đ, đ)âđŠ + đ(đ, đ)
Note: This can be extended to any number variables. In three variables we have:
đ(đ + âđ„, đ + âđŠ, đ + âđ§) â đđ„(đ, đ, đ)âđ„ + đđŠ(đ, đ, đ)âđŠ + đđ§(đ, đ, đ)âđ§ + đ(đ, đ)
If âđ„ and âđŠ are sufficiently small, then we can approximate âđ as
âđ â đđ„(đ, đ)âđ„ + đđŠ(đ, đ)âđŠ
The differential of đ(đ„, đŠ) is defined as
đđ = đđ„(đ„, đŠ)đđ„ + đ
đŠ(đ„, đŠ)đđŠ
= đđ
đđ„đđ„ +
đđ
đđŠđđŠ
Multivariable Chain Rule
Let đ(đ„1, . . . , đ„đ) be a differentiable function of đ variables. Suppose that each of the variables, đ„1, . . . , đ„đ, is a differentiable function of đ independent variables, đĄ1, . . . , đĄđ. Then for đ = 1, ⊠,đ
đđ
đđĄđ=
đđ
đđ„1
đđ„1
đđĄđ+
đđ
đđ„2
đđ„2
đđĄđ+. . . +
đđ
đđ„đ
đđ„đ
đđĄđ
Note: Since đ„đ is assumed to be a function of more than one variable, the partial derivative notation is
required, đđ„đ
đđĄđ. If đ = 1 then
đđ„đ
đđĄ could be used.
Multivariable Implicit Differentiation
Suppose we have the equation đč(đ„, đŠ) = 0, and that đč(đ„, đŠ) is differentiable. Then đđŠ
đđ„= â
đčđ„đčđŠ
Provided đčđŠ â 0
Suppose we have the equation đč(đ„, đŠ, đ§) = 0, and that đč(đ„, đŠ, đ§) is differentiable. Then
đđ§
đđ„= â
đčđ„đčđ§
and đđ§
đđ„= â
đčđ„đčđ§
Provided đčđ§ â 0
The Directional Derivative
Let đ(đ„, đŠ) be a function of two variables and let đ denote a unit vector. Then the derivative of đ(đ„, đŠ) in the direction of đ is called the directional derivative, đ·đđ.
đ·đđ = đ»đ â đ Where,
đ»đ = âšđđ
đđ„,đđ
đđŠâ© and đ = âšđąđ„, đąđŠâ©
The definition can be extended to three or more dimensions as follows where
đ»đ = âšđđ
đđ„1, . . . ,
đđ
đđ„đ
â© and đ = âšđąđ„1, . . . , đąđ„đ
â©
Algebraic Properties of the Gradient Vector
If đ(đ„, đŠ, đ§) and đ(đ„, đŠ, đ§) are differentiable functions and đ is a constant, then i. đ»(đ + đ) = đ»đ + đ»đ
ii. đ»(đđ) = đđ»đ
iii. Product Rule for Gradients: đ»(đđ) = đ»đđ + đđ»đ
iv. Chain Rule for Gradients: If đč(đĄ) is a differentiable function of one variable, then
đ» (đč(đ(đ„, đŠ, đ§))) = đčâČ(đ(đ„, đŠ, đ§))đ»đ
Gradient Vector as the Direction of Maximum Increase
Let đ be a differentiable function at a fixed point, đ, with đ»đ|đ â 0.
âą đ»đ points in the direction of the maximum rate of increase of đ at đ, and the maximum rate of increase is âđ»đâ.
âą âđ»đ points in the direction of the maximum rate of decrease of đ at đ, and the maximum rate of decrease is âđ»đâ.
Gradient Vector as a Normal Vector
Let đ be a point on a level curve, đ(đ„, đŠ) = đ, or on a level surface, đ(đ„, đŠ, đ§) = đ, and assume that đ»đ|đ â 0. Then đ»đ|đ is a vector that is normal to the tangent line/plane to the curve/surface at the point đ. Moreover, the tangent line/plane to the curve/surface at the point đ has the equation
Tangent Line
: đđ„(đ„0, đŠ0)(đ„ â đ„0) + đđŠ(đ„0, đŠ0)(đŠ â đŠ0) = 0
Tangent Plane
: đđ„(đ„0, đŠ0, đ§0)(đ„ â đ„0) + đđŠ(đ„0, đŠ0, đ§0)(đŠ â đŠ0) + đđ§(đ„0, đŠ0, đ§0)(đ§ â đ§0) = 0
Critical Points Definition for Two Variable Functions
A point đ = (đ, đ) in the domain of đ(đ„, đŠ) is called a critical point if:
âą đđ„(đ, đ) = 0 or đđ„(đ, đ) does not exists, AND
âą đđŠ(đ, đ) = 0 or đđŠ(đ, đ) does not exists.
Fermatâs Theorem of Local Extrema for Two Variable Functions
If đ(đ, đ) is a local minimum or maximum, then đ = (đ, đ) is a critical point of đ(đ„, đŠ). Note: This theorem does not claim that all critical points are local extreme values, but rather that all local extreme values are critical points.
Second Derivative Test for Two Variable Functions
Let đ = (đ, đ) be a critical point of the function, đ(đ„, đŠ) and assume đđ„đ„, đđŠđŠ and đđ„đŠ are
continuous near đ. Then: 1. If đ· > 0 and đđ„đ„(đ, đ) > 0, then đ(đ, đ) is a local minimum. 2. If đ· > 0 and đđ„đ„(đ, đ) < 0, then đ(đ, đ) is a local maximum. 3. If đ· < 0 then đ(đ, đ) is a saddle point. 4. If đ· = 0 then the test is inconclusive. Where đ· is called the discriminate
đ· = đđ„đ„(đ, đ)đđŠđŠ(đ, đ) â đđ„đŠ2(đ, đ)
Existence and Location of Absolute Extrema
Let đ(đ„, đŠ) be a continuous function on a closed domain đ· in đ 2. Then: 1. đ(đ„, đŠ) takes on both a minimum and maximum value on đ·. 2. The extreme values occur either at critical points in the interior of đ· or at points on the
boundary of đ·.
Optimizing with Constraints
Optimizing with constraints involves finding the minimum or maximum value of a function, e.g. đ(đ„1, . . . , đ„đ) subject to the fact that the independent variables are related in some fashion, e.g. đ(đ„1, . . . , đ„đ) = 0. The terminology used is as follows:
Objective Function đ(đ„1, . . . , đ„đ)
Expresses the quantity we would like to optimize in terms of đ independent variables.
Constraint Function đ(đ„1, . . . , đ„đ) = 0
Expresses a relationship between the independent that must be satisfied within the context of optimizing the objective function.
Lagrange Multiplier Theorem
Assume đ(đ„, đŠ) and đ(đ„, đŠ) are differentiable functions. If đ(đ„, đŠ) has a local extremum on the constraint curve, đ(đ„, đŠ) = 0, at đ = (đ, đ) and if đ»đđ â 0, then there is a scalar, đ, such that
đ»đđ = đđ»đđ
Lagrange Multipliers Technique Applied to Optimization with Constraints
The above Lagrange Multiplier Theorem can be applied to optimization problems with constraints. The theorem can be generalized to any number of variables and any number of constraints functions as follows: Given an đ variable differentiable objective function, đ(đ„1, . . . , đ„đ), and đ differentiable constraint functions, {đ1(đ„1, . . . , đ„đ) = 0, . . . , đđ(đ„1, . . . , đ„đ) = 0}. The Lagrange condition is written as follows:
đ»đđ = âđđđ»đđ,đ
đ
đ=1
Expanding this expression creates đ equations that can then be used to find the extreme values of đ(đ„1, . . . , đ„đ) subject to {đ1(đ„1, . . . , đ„đ) = 0, . . . , đđ(đ„1, . . . , đ„đ) = 0}.
Next series of lessons focused on integration of multivariable scalar functions. Although vectors are not prominent in these lessons, multivariable integration is used to compute many different quantities on science and engineering. Multivariable integration is a natural extension of single variable integration. Alternate coordinate systems also become important is some multivariable integration problems.
Double Integral over a Rectangular Region
The definite double integral of đ(đ„, đŠ) over a rectangular region, đ , is the limit of the Riemann Sum.
âŹđ(đ„, đŠ)đđŽ
đ
= đđđâđââ0
{âđ(đ„đ, đŠđ)âđŽđ
đ
đ=1
} = đđđâđââ0
{âđ(đ„đ, đŠđ)âđ„đâđŠđ
đ
đ=1
}
When this limit exists, we say đ(đ„, đŠ) is integrable over đ .
* * *
* *
* * *
*
x
y đ(đ„đ , đŠđ)
âđ„đ
âđŠđ *
âđŽđ
Fubiniâs Theorem
The double integral of a continuous function đ(đ„, đŠ) over the rectangular region, đ = {(đ„, đŠ)|đ †đ„ †đ, đ †đŠ †đ}, is equal to the iterated single integral (in either order).
âŹđ(đ„, đŠ)đđŽ
đ
= â« â« đ(đ„, đŠ)đđ„đ
đ
đđŠđ
đ
= â« â« đ(đ„, đŠ)đđŠđ
đ
đđ„đ
đ
When đ(đ„, đŠ) = đ(đ„)â(đŠ), the double integral can be expressed as the product of two integrals as shown below.
â« â« đ(đ„, đŠ)đđ„đ
đ
đđŠđ
đ
= (â« đ(đ„)đđ„đ
đ
)(â« â(đŠ)đđŠđ
đ
)
Double Integral over Vertically Simple Regions
A vertically simple region is defined as:
đ· = (đ„, đŠ)| đ †đ„ †đ, đ1(đ„) †đŠ †đ2(đ„)
And the double integral of đ(đ„, đŠ) over đ· is
âŹđ(đ„, đŠ)đđŽ
đ·
= â« â« đ(đ„, đŠ)đđŠđđ„đ2(đ„)
đ1(đ„)
đ
đ
y
x
a b
c
d
D
g1(x)
g2(x)
Double Integral over Horizontally Simple Regions
A horizontally simple region is defined as:
đ· = {(đ„, đŠ)| đ1(đŠ) †đ„ †đ2(đŠ), đ †đŠ †đ} And the double integral of đ(đ„, đŠ) over đ· is
âŹđ(đ„, đŠ)đđŽ
đ·
= â« â« đ(đ„, đŠ)đđ„đ2(đŠ)
đ1(đŠ)
đđŠđ
đ
y
x
c
d
D
g1(y)g2(y)
Volume Between Two Surfaces
Assuming the integrable functions, đ1(đ„, đŠ) â„ đ2(đ„, đŠ), for all points in đ·, then the volume between the surfaces is given as
đ = âŹ( đ1(đ„, đŠ) â đ2(đ„, đŠ))đđŽ
đ·
Triple Integral Over a Boxed Region
The triple integral of a continuous function đ(đ„, đŠ, đ§) over a box, đ is:
âđ(đ„, đŠ, đ§)đđ
đ
= â« â« â« đ(đ„, đŠ, đ§)đđ§đđŠđđ„đ
đ§=đ
đ
đŠ=đ
đ
đ„=đ
Where,
đ = (đ„, đŠ, đ§)| đ †đ„ †đ, đ †đŠ †đ, đ †đ§ †đ
Furthermore, the integral can be evaluated in any order.
Triple Integral Over a General Region
For a general region, the triple integral is best written as follows:
âđ§đđ
đ·
= âŹ(â« đ(đ„, đŠ, đ§)đđđ2
đ1
)
đ
đđŽ
Where, the inner integral is with respect to any one of the three variables, i.e. we choose đ to be one of the elements of the set {đ„, đŠ, đ§}. The region, đ , is the projection of the solid object in the plane defined by the two remaining variables. We can then express the đđŽ in two different orders for each of the 3 possible projections as shown below.
Area and Volume
The area of a general region can be found using the double integral of đ(đ„, đŠ) = 1 over a region, đ . For example
đ = {(đ„, đŠ)| đ †đ„ †đ, đ †đŠ †đ}
âŹ1đđŽ
đ
= â« â« 1đđ„đđŠđ
đ
đ
đ
= (đ â đ) â (đ â đ) = đŽđđđ đđ đ đđđđđ
The volume of a general region can be found using the triple integral of đ(đ„, đŠ, đ§) = 1 over a region, đ . For example
đ = {(đ„, đŠ, đ§)| đ †đ„ †đ, đ †đŠ †đ, đ †đ§ †đ}
â1đđ
đ
= â« â« â« 1đđ„đđŠđđ§đ
đ
=đ
đ
đ
đ
(đ â đ) â (đ â đ) â (đ â đ) = đđđđąđđ đđ đ đđđđđ
Double Integral in Polar Coordinates
For a continuous function, đ, on the domain, đ· = {(đ, đ)| đ1 †đ †đ2, đ1 †đ †đ2}
âŹđ(đ„, đŠ)đđŽ
đ·
= â« â« đ(đ đđđ (đ) , đ đ đđ(đ))đ2
đ=đ1
đ2
đ=đ1
đđđđđ
đđŽ
đ
đ
đđ
đđ
đđđ
đ„
đŠ
đđŽ = đđđđđ
Triple Integral in Cylindrical Coordinates
For a continuous function, đ, on the domain, đ· = {(đ, đ, đ§)| đ1 †đ †đ2, đ1 †đ †đ2, đ§1 â€đ§ †đ§2 }
âđ(đ„, đŠ, đ§)đđ
đ·
= â« â« â« đ(đ đđđ (đ) , đ đ đđ(đ) , đ§)đ§2
đ§=đ§1
đ2
đ=đ1
đđđ§đđđđđ2
đ=đ1
z
y
xđđđ
đđ
đđ đđ§
đđ = đđđ§đđđđ
Triple Integral in Spherical Coordinates
For a continuous function, đ, on the domain, đ· = {(đ, đ, đ)| đ1 †đ †đ2, đ1 †đ â€đ2, đ1 †đ †đ2, }
âđ(đ„, đŠ, đ§)đđ
đ·
= â« â« â« đ(đ đ đđ(đ) đđđ (đ) , đ đ đđ(đ) đ đđ(đ) , đ đđđ (đ))đ2
đ=đ1
đ2
đ=đ1
đ2 đ đđ(đ) đđđđđđđ2
đ=đ1
y
x
z
đ
đ đ
đ đđ
đđ đđđ
đđ
đđ
đđđ
đđ = đ2 đ đđ(đ) đđđđđđ
Total Amount Using Density
One Dimension
đđđĄđđ đŽđđđąđđĄ = â« đż(đ„)đđ„
đ
Where, đż(đ„) is the amount per unit length and đ is the interval of integration
Two Dimensions
đđđĄđđ đŽđđđąđđĄ = âŹđż(đ„, đŠ)đđŽ
đ
Where, đż(đ„, đŠ) is the amount per unit area and đ is the region of integration.
Three Dimensions
đđđĄđđ đŽđđđąđđĄ = âđż(đ„, đŠ, đ§)đđ
đ
Where, đż(đ„, đŠ, đ§) is the amount per unit volume and đ is the region of integration.
Center of Mass
One Dimension
đ„đđđ =â« đ„đż(đ„)đđ„đ
â« đż(đ„)đđ„đ
Where, đż(đ„) is mass density per unit length and đ is the interval of integration
Two Dimensions
đ„đđđ =⏠đ„đż(đ„, đŠ)đđŽ
đ
⏠đż(đ„, đŠ)đđŽđ
đŠđđđ =⏠đŠđż(đ„, đŠ)đđŽ
đ
⏠đż(đ„, đŠ)đđŽđ
Where, đż(đ„, đŠ) is the mass density per unit area and đ is the region of integration.
Three Dimensions
đ„đđđ =â đ„đż(đ„, đŠ)đđ
đ
â đż(đ„, đŠ)đđđ
đŠđđđ =â đŠđż(đ„, đŠ)đđ
đ
â đż(đ„, đŠ)đđđ
đ§đđđ =â đ§đż(đ„, đŠ)đđ
đ
â đż(đ„, đŠ)đđđ
Where, đż(đ„, đŠ, đ§) is the mass density per unit volume and đ is the region of integration.
Rotational Inertia (2nd Moment)
One Dimension
đŒ = â« đ„2đż(đ„)đđ„
đ
Where, đż(đ„) is mass density per unit length and đ is the interval of integration
Two Dimensions
đŒđ„ = âŹđ„2đż(đ„, đŠ)đđŽ
đ
đŒđŠ = âŹđŠ2đż(đ„, đŠ)đđŽ
đ
đŒđ§ = âŹ(đ„2 + đŠ2)đż(đ„, đŠ)đđŽ
đ
Where, đż(đ„, đŠ) is the mass density per unit area, đ is the region of integration, and đŒđ„,đŠ,đ§ is
the rotational inertia with respect to the đ„, đŠ, đ§-axis respectively.
Three Dimensions
đŒđ„
= â(đŠ2 + đ§2)đż(đ„, đŠ, đ§)đđ
đ
đŒđŠ
= â(đ„2 + đ§2)đż(đ„, đŠ, đ§)đđ
đ
đŒđ§
= â(đ„2 + đŠ2)đż(đ„, đŠ, đ§)đđ
đ
Where, đż(đ„, đŠ, đ§) is the mass density per unit volume, đ is the region of integration, and đŒđ„,đŠ,đ§ is the rotational inertia with respect to the đ„, đŠ, đ§-axis respectively.
Probability Density Functions
One Random Variable
đ(đ †đ †đ) = â« đ(đ„)đđ„đ
đ„=đ
Two Random Variables
đ(đ †đ †đ; đ †đ †đ) = â« â« đ(đ„, đŠ)đđ„đđŠđ
đ„=đ
đ
đŠ=đ
Three Random Variables
đ(đ †đ †đ; đ †đ †đ; đ †đ †đ) = â« â« â« đ(đ„, đŠ, đ§)đđ„đđŠđđ§đ
đ„=đ
đ
đŠ=đ
đ
đ§=đ
The Jacobian Determinant
Given the transformation đ: đ 2 â đ 2, where đ is defined as
đ(đą, đŁ) = (đ„(đą, đŁ), đŠ(đą, đŁ))
The Jacobian of đ, đœđđ(đ), is given as
đœđđ(đ) = |
đđ„
đđą
đđ„
đđŁđđŠ
đđą
đđŠ
đđŁ
| =đđ„
đđąâđđŠ
đđŁâ
đđ„
đđŁâđđŠ
đđą
The Jacobian generalizes to đ dimensions. For example, with three variables we have đ: đ 3 â đ 3, where đ is defined as
đ(đą, đŁ, đ€) = (đ„(đą, đŁ, đ€), đŠ(đą, đŁ, đ€), đ§(đą, đŁ, đ€))
đœđđ(đ) =đ(đ„, đŠ, đ§)
đ(đą, đŁ, đ€)=
|
|
đđ„
đđą
đđ„
đđŁ
đđ„
đđ€đđŠ
đđą
đđŠ
đđŁ
đđŠ
đđ€đđ§
đđą
đđ§
đđŁ
đđ§
đđ€
|
|
âą The Jacobian of đ is also denoted as đ(đ„,đŠ)
đ(đą,đŁ),đ(đ„,đŠ,đ§)
đ(đą,đŁ,đ€)
âą The Jacobian is sometimes meant to express the matrix only and not its determinant. In these cases, we refer to the above as the Jacobian Determinant.
Change of Variable Formula in
Let đ: (đą, đŁ) â (đ„, đŠ) be a mapping from đą-đŁ space to đ„-đŠ space that is one-to-one. If đ(đ„, đŠ) is continuous, then
âŹđ(đ„, đŠ)đđ„đđŠ
đ·
= âŹđ(đ„(đą, đŁ), đŠ(đą, đŁ)) |đ(đ„, đŠ)
đ(đą, đŁ)| đđąđđŁ
đ
Where, đ· is some region in đ„-đŠ space and đ is the corresponding region in đą-đŁ space. Note: The Change of Variables Formula as stated above turns an đ„đŠ integral into a đąđŁ integral, but the
map, đ, goes from the đąđŁ domain to the đ„đŠ domain, i.e. đ(đą, đŁ) = (đ„(đą, đŁ), đŠ(đą, đŁ))
In đ 3 we have:
âđ(đ„, đŠ, đ§)đđ„đđŠđđ§
đ·
= âđ(đ„(đą, đŁ, đ€), đŠ(đą, đŁ, đ€), đ§(đą, đŁ, đ€)) |đ(đ„, đŠ, đ§)
đ(đą, đŁ, đ€)| đđąđđŁđđ€
đ
The next series of lessons pushed integration even further to includes integration over curves and surfaces. In addition, we introduced integration over vector fields.
Vector Field
A vector field is a function that assigns a vector to each point, đ = âšđ„, đŠ, đ§â©, in space. In three dimensions it is denoted as
đ(đ„, đŠ, đ§) = âšđč1(đ„, đŠ, đ§), đč2(đ„, đŠ, đ§), đč3(đ„, đŠ, đ§)â©
A unit vector field, đđč, is defined as
đđč =đ(đ„, đŠ, đ§)
âđ(đ„, đŠ, đ§)â
An important example is a unit radial vector.
Two dimensional unit radial vector Three dimensional unit radial vector
đđ = âšđ„
đ,đŠ
đâ©
Where, đ = âđ„2 + đŠ2
đđ = âšđ„
đ,đŠ
đ,đ§
đâ©
Where, đ = âđ„2 + đŠ2 + đ§2
Divergence of a Vector Field
The divergence of a vector field, đ, results in a scalar function. In three dimensions it is defined as
đđđŁ(đ) = đ» â đ = âšđ
đđ„,đ
đđŠ,đ
đđ§ â© â âšđč1, đč2, đč3â© =
đđč1
đđ„+
đđč2
đđŠ+
đđč3
đđ§
The divergence generalizes to an arbitrary number of dimensions.
Divergence Intuition â Assume đ is a fluid velocity vector field
The divergence of a vector field represents the degree to which the fluid is flowing in towards or away from each point in space.
đđđŁ(đ(0,0)) > 0 đđđŁ(đ(0,0)) < 0 đđđŁ(đ) = 0
x
y
x
y
x
y
There is a net liquid flow outward from the origin.
There is a net liquid flow inward from the origin.
There is a net flow of zero at any point in space.
Curl of a Vector Field
The curl of a vector field, đ, results in a vector function. It is defined as
đđąđđ(đ) = đ» Ă đ = ||
ïżœÌïżœ đÌ ïżœÌïżœđ
đđ„
đ
đđŠ
đ
đđ§
đč1 đč2 đč3
|| = âš(đđč3
đđŠâ
đđč2
đđ§) , (
đđč1
đđ§â
đđč3
đđ„) , (
đđč2
đđ„â
đđč1
đđŠ) â©
The curl is defined on three dimensions only.
Curl Intuition â Assume đ is a fluid velocity vector field
The curl measures the amount to which the fluid circulates around a fixed axis at each point in space.
P
P
P
The paddle wheel rotates counterclockwise and the resulting vector points out of the page.
The paddle wheel rotates clockwise and the resulting vector points into the page.
The paddle wheel does not rotate.
Conservative Vector Fields
âą If đ = đ»đ, then đ is called the potential function for đ.
âą đ is called conservative it has a potential function.
âą Potential functions are unique up to a constant, đ¶. The vector field, đ, is conservative if
đđąđđ(đ) = 0 Or equivalently,
đđč3
đđŠ=
đđč2
đđ§,
đđč3
đđ„=
đđč1
đđ§,
đđč2
đđ„=
đđč1
đđŠ
Scalar Line Integral
The scalar line integral of the function đ(đ„, đŠ, đ§) over the curve, đ¶, is given as
â« đ(đ„, đŠ, đ§)đđ
đ¶
Let đ(đĄ) be a parameterization of a curve, đ¶, for đ †đĄ †đ, then the scalar line integral is also given as
â« đ(đ„, đŠ, đ§)đđ
đ¶
= â« đ(đ(đĄ))âđâČ(đĄ)âđđĄđ
đ
Vector Line Integral
The vector line integral of the vector field đ(đ„, đŠ, đ§) over the curve, đ¶, is given as
â« đ(đ„, đŠ, đ§) â đđ
đ¶
Let đ(đĄ) be a parameterization of a curve, đ¶, for đ †đĄ †đ, then the vector line integral is also given as by the two equivalent expressions
â« (đ(đ(đĄ)) â đâČ(đĄ)) đđĄđ
đ
= â« (đ1(đ(đĄ))đđ„
đđĄ+ đ2(đ(đĄ))
đđŠ
đđĄ+ đ3(đ(đĄ))
đđ§
đđĄ) đđĄ
đ
đ
Work done by a Vector Force Field
The work done on a particle moving along curve parameterized by đ(đĄ) in the presence of a vector force field, đ, is given as
đ = â« (đ(đ(đĄ)) â đâČ(đĄ)) đđĄđ
đ
Flux Across a Plane Curve
The flux across a plane curve parameterized by đ(đĄ) in the presence of a vector field, đ, is given as
Ί = â« (đ(đ(đĄ)) â đ”(đĄ)) đđĄđ
đ
Where, đ”(đĄ) = âšđŠâČ(đĄ), âđ„âČ(đĄ)â© and đâČ(đĄ) = âšđ„âČ(đĄ), đŠâČ(đĄ)â©
The Fundamental Theorem for Conservative Vector Fields
Assume đ = đ»đ on a domain đ·. 1. If đ is a path along a curve đ¶ from đŽ to đ” in đ·, then
â« đ â đđđ¶
= â«đ
đđĄđ(đ(đĄ))
đ
đ
đđĄ = đ(đ(đ)) â đ(đ(đ)) = đ(đ”) â đ(đŽ)
In other words, đ is path-independent 2. The circulation around a closed curve đ¶, (i.e. đŽ = đ”) is zero
âź đ â đđđ¶
= 0
Conservative Vector Field Criteria
The vector field đ is conservative on a simply connected domain, đ·, if đ satisfies the cross-
partials conditions derived from the fact that the curl is zero.
đđąđđ(đ) = 0 â đđč3
đđŠ=
đđč2
đđ§,
đđč3
đđ„=
đđč1
đđ§,
đđč2
đđ„=
đđč1
đđŠ
Conservative Fields in Physics
The gravitational force and the electrostatic forces are conservative forces. They are both governed by the inverse square law.
Inverse Square Law Force and its Potential Function
đđȘ(đ„, đŠ, đ§) =đ¶
đ2đđ â đ(đ„, đŠ, đ§) =
đ¶
đ
Specifically,
Gravitational Force Electrostatic Force
đđș = âđșđđ
đ2đđ đđŹ =
đđđ
đ2đđ
Where, đș = 6.67đžâ11, đ = 8.9đž9, đ1 and đ2 are the two masses in kilograms, đ1 and đ2
are the two charges in Coulombs, and đđ = âšđ„
đ,đŠ
đ,đ§
đâ©.
Scalar Line Integral
Let đ(đĄ) be a parameterization of a curve, đ¶, for đ †đĄ †đ, then the scalar line integral is also given as
â« đ(đ„, đŠ, đ§)đđ
đ¶
= â« đ(đ(đĄ))âđâČ(đĄ)âđđĄđ
đ
Vector Line Integral
Let đ(đĄ) be a parameterization of a curve, đ¶, for đ †đĄ †đ, then the vector line integral is also given as by the two equivalent expressions
â« đ(đ„, đŠ, đ§) â đđ
đ¶
â« (đ(đ(đĄ)) â đâČ(đĄ)) đđĄđ
đ
Work Along a Curve Flux Across a Curve
đ = â« (đ(đ(đĄ)) â đâČ(đĄ)) đđĄđ
đ
Ί = â« (đ(đ(đĄ)) â đ”(đĄ)) đđĄđ
đ
Scalar Surface Integral
Let đź(đą, đŁ) be a parameterization of a surface, đź, on the domain. The scalar surface integral of the function đ(đ„, đŠ, đ§) over the surface on the given domain is
⏠đ(đ„, đŠ, đ§)đđđź
= ⏠đ(đź(đą, đŁ))đ·
âđ”(đą, đŁ)âđđąđđŁ
For đ(đ„, đŠ, đ§) = 1, we obtain the surface area on the domain đ·.
đŽđđđ(đź) = ⏠âđ”(đą, đŁ)âđđąđđŁđ·
Scalar Surface Integral over a Surface đ = đ(đ, đ)
The scalar surface integral of the function đ(đ„, đŠ, đ§) over a portion of a surface that can be represented as đ§ = đ(đ„, đŠ), is given as
⏠đ(đ„, đŠ, đ§)đđđź
= ⏠đ(đ„, đŠ, đ(đ„, đŠ))đ·
(âđđ„2 + đđŠ
2 + 1)đđ„đđŠ
Vector Surface Integral
Let đź(đą, đŁ) be a parameterization of a surface, đź, on the domain, đ·. The vector surface integral, also called the flux, of the vector field đ(đ„, đŠ, đ§) over the surface on the given domain is
⏠(đ â đ)đđđź
= ⏠(đ(đź(đą, đŁ)) â đ”(đą, đŁ)) đđąđđŁđ·
The final series of lessons introduced the Fundamental Theorems of Vector Calculus. These theorems are also shown to follow directly from the Fundamental Theorem of Single Variable Calculus. These theorems are extremely important and can be used as a gateway for more advanced studies in various science and engineering applications.
Greenâs Theorem
Let đ· be a domain in đ 2 whose boundary is a simple closed curve, đ¶, oriented
counterclockwise. Then
⏠(đđąđđđ§(đ ))đđŽđ·
= âź đ â đđđ¶
Where, đđąđđđ§(đ) = (đđč2(đ„,đŠ)
đđ„â
đđč1(đ„,đŠ)
đđŠ)
With đ = âšđč1(đ„, đŠ), đč2(đ„, đŠ)â© and đđ = âšđđ„, đđŠâ©, we can also express the line integral as
âź đ â đđđ¶
= âź đč1(đ„, đŠ)đđ„ + đč2(đ„, đŠ)đđŠđ¶
Area of Region Using Greenâs Theorem
There are three equivalent formulas we can use for the area of a region, đ·, enclosed by a
simple curve, đ¶.
đŽđđđ đžđđđđđ đđ đđŠ đ¶ = (âź đ„đđŠđ¶
) = (âź âđŠđđ„đ¶
) = (1
2âź đ„đđŠ â đŠđđ„đ¶
)
Greenâs Theorem Using Normal Vector â Flux
Using the normal vector to the curve, Greenâs Theorem can be used to express the flux across the curve as follows
⏠đđđŁ(đ)đđŽđ·
= âź (đ â đ”)đđĄđ¶
Where, đ”(đĄ) = âšđŠâČ(đĄ), âđ„âČ(đĄ)â© and đđđŁ(đ) = (đđč1
đđ„+
đđč2
đđŠ)
General Form of Greenâs Theorem
Greenâs theorem can also be applied to non-simple regions as long as we keep in mind the fact that the region to be considered always lies to the left of the curve according to its orientation.
C1
C2
D
C3
In this example the region, D, is represented as
đ· = đ¶1 + đ¶2 â đ¶3
Surfaces and Surface Boundaries
Different surfaces may have different types of boundaries. For example, the surface below has a single simple closed curve as its boundary. We define the orientation of the curve as follows:
âą When you walk around the curve with your body pointing out in the direction of the normal vector, you should be walking in such a way that the surface is to your left side.
n
C
S
y
x
z
Stokesâ Theorem
Let đ be an oriented smooth surface that is bounded by a single simple closed curve, đ¶, and let đ be a vector field. Then
⏠đđąđđ(đ) â đđșđ
= âź đ â đđđ¶
Where, đđąđđ(đ) = đ» Ă đ
Surface Independence
The surface integral of a vector field, đ, with an associated vector potential function, đš, (where đ = đđąđđ(đš)), is surface independent. It depends only on the boundary curve, đ¶.
⏠đ â đđșđđđ„
= ⏠đđąđđ(đš) â đđșđđđ„
= âź đš â đđđ¶
Divergence Theorem
Let đ be a closed surface that encloses a region, đ, in đ 3. Assume that S is piecewise smooth and is oriented by a normal vector pointing to the outside of đ. Let đ be a vector field whose domain contains đ. Then
â đđđŁ(đ)đđđ
= ⏠đ â đđșđ
Gaussâs Law
The electric flux through a closed surface is proportional to the total charge enclosed within the surface.
⏠đŹ â đđșđ
=đđ
đ0
Where, đŹ = (đ
4đđ0) (
ïżœÌïżœ
đ2) and ïżœÌïżœ = âšđ„
đ,đŠ
đ,đ§
đâ©.
Relationship Between Fundamental Theorems
In a general sense the theorems relate the integral of some type of derivative of some function over some region to the values of that function along the boundary of the region.
Fundamental Theorem of Single Variable Calculus
â« đâČ(đĄ)đđĄ đ
đ
= đ(đ) â đ(đ)
Relates the integral of the derivative of a scalar function over a one-dimensional region to the values of the function at the endpoints of the region.
Gradient Theorem
â« đ»đ â đđđ¶
= đ(đ) â đ(đ)
Relates the integral of the gradient of a scalar function over a curve in three dimensions, đ¶, to the values of that function at the endpoints of the curve.
Greenâs Theorem
⏠đđąđđđ§(đ)đđŽđ·
= âź đ â đđđ¶
Relates the double integral of the two dimensional curl of a vector field over a region, đ·, to the value of the line integral of that vector field along the boundary curve for that region, đ¶.
Stokesâ Theorem
⏠đđąđđ(đ) â đđșđ
= âź đ â đđđ¶
Relates the surface integral of the three dimensional curl of a vector field over a surface, đ, to the value of the line integral of that vector field along the boundary curve for that surface, đ¶.
Divergence Theorem
â đđđŁ(đ)đđđ
= ⏠đ â đđșđ
Relates the triple integral of the divergence of a vector field over a 3D region, đ, to the value of the surface integral of that vector field over the boundary surface for the region, đ.
By: ferrantetutoring
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