13.1_13.2_matb_113_notes_e.doc
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16 PARTIAL DIFFERENTIATION
13. VECTOR-VALUED FUNCTIONS
AND MOTION IN SPACE 13.1 Curves in Space and Their TangentsA vector-valued function r is represented by
where and are scalar functions.
Illustration on vector valued function Example:1 Graphing a vector valued function
(i) Sketch .
(ii) Sketch .Limits and Continuity
Definitions Limit of Vector functions
Let be a vector function and L a vector, we say that r has limit L as t approaches to and write
if, for every number , there is a corresponding number such that for all t
Example:2
Find the .
Definition: Continuous at a PointA vector function is continuous at a point in its domain if
.The function is continuous if it is continuous at every pointing its domain.
Example:3
Show that the function is discontinuous at a certain point t.
Derivatives and Motion
Definition Derivative
The vector function has a derivative (is differentiable) at t if
f, g and h have derivatives at t. The derivative is the vector function
A curve that is made up of a finite number of smooth curves pieced together in a continuous fashion is called piecewise smooth.
Definitions: Velocity, direction, Speed,
Acceleration
If r is the position vector of a particular moving along a smooth curve in space, then
is the particles velocity vector, tangent to the curve. At any time t, the direction of
v is the direction of motion, the magnitude of v is the particles speed, and the derivative , when it exists, is the particles acceleration vector. In summary
1. Velocity is the derivative of position:
2. Speed is the magnitude of the velocity:
Speed =
3. Acceleration is the derivative of
velocity:
4. The unit vector is the direction of
motion at time t.
Example:4
The position of a particle in space at time t is given by
.
Find (a) the velocity and acceleration vector at t = 2.
(b) the speed at any time t.
(c) the time t, if any when the particles acceleration is
orthogonal to its velocity.
Differentiation Rules for Vector Functions
Let u and v be differentiable vector functions of t, C a constant vector, c any scalar, and f any differentiable scalar function.
1. Constant function Rule:
2. Scalar multiple Rule:
3. Sum rule:
4. Difference Rule:
5. Dot Product Rule:
6. Cross Product Rule:
7. Chain Rule:
Proof of the Dot Product Rule
Proof of the Chain Rule
Vector Functions of Constant Length
If r is a differentiable vector function of t of constant
Length, then
Example:5
Show that has constant length and is orthogonal to its derivative.
13.2 Integrals of Vector Functions; Projectile
MotionDefinition Indefinite Integral
The indefinite integral of r with respect to t is the set of all antiderivatives of r denoted by . If R is any antiderivative of r, then
Example:6
Evaluate .Definition Definite Integral
If the components of
are integrals over ,
then so is r, and the definite integral of r from a to b is
Example: 7
Evaluate .Example: 8
Find the vector valued function r(t) given by
with initial conditions and
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