functions of several variables2

8/13/2019 Functions of Several Variables2

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(x0, y0, z0)

−→n = a,b,c

a(x − x0) + b(y − y0) + c(z − z0) = 0

ax + by + cz = d

−→n

−→n

P (2,−1, 3) Q(1, 4, 0) R(0,−1, 5)

−−→P Q = −1, 5,−3

−→P R = −2, 0, 2

−−→P Q ×

−→P R =

i ˆ j k

−1 5 −3−2 0 2

= 10, 8, 10

10(x − 2) + 8(y − (−1)) + 10(z − 3) = 0

5x + 4y + 5z = 21

cos(θ)

Q : x + 2y + z = 5 R : 2x + y − z = 7

z = 0 x y x = 3 y = 1 z = 0

−→v = −→n 1 ×−→n 2)

P

−−→P Q Q

−→n scal−→n−−→P Q



P t = 0 Q

−→v

−−→P Q

−→v |−→v |

z = x2 R

3

f (x, y) =

9 − x2 − y2

D = {(x, y) ∈ R2|x2 + y2 ≤ 9}

R = [0, 3]

f (x, y) = c

c

f (x, y) =

9 − x2 − y2 = c

x2 + y2 = 9 − c2 ∴ c

∂

z = 2x sin(y) ∂z

∂x = 2sin(y)

∂ 2f

∂x2 = f xx =

∂

∂x(f x)

(a, b)

f xy = f yx

dy

dx= −

F x

F y



∂z

∂x x3 + y3 + z3 + 6xyz = 1 z x y

3x2 + 3z2∂z

∂x + (6yz + 6xy

∂z

∂x) = 0 ← z x

∂z

∂x = −3x2 − 6yz

3z2 + 6xy

−→∇f = f x, f y, f z

D−→u f = −→∇f · −→u

−→u a,b,c

−→u

|−→u |

−→∇f · −→u = |

−→∇f ||−→u | cos(θ) θ = 0◦

−→u

u =

−→∇f

|−→∇f |

∴ u

|−→∇f |

x y

−→∇ ⊥

F (x ,y ,z) = 0 F x(x0, y0, z0)(x − x0) + F y(x0, y0, z0)(y − y0) + F z(x0, y0, z0)(z − z0) = F P 0(x0, y0, z0) z = F (x, y)

z = f (x, y) : z = F x(x0, y0)(x − x0) + F y(x0, y0)(y − y0) + F (x0, y0)

L(x, y) = F x(x0, y0)(x − x0) + F y(x0, y0)(y − y0) + F (x0, y0)

dz = F x(x, y)dx + F y(x, y)dy

1) f x(a, b) = 0 andf y(a, b) = 0

±

2) F x or F y DN E

D(a, b) = f xx(a, b)f yy(a, b) − (f xy(a, b))2

D > 0 f xx(a, b) > 0 f (a, b)

D > 0 f xx(a, b) < 0 f (a, b)

D < 0 f (a, b)

D = 0

f D f

D



f D

f

x2 + y2 ≤ 25 x = a cos(θ) y = a sin(θ) a

x

y

(x ,y ,z) λ

−→∇f (x ,y ,z) = λ

−→∇g(x ,y ,z)

λ x ,y ,z

functions of several variables2

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