Chapter 2 - Chain Rules

Question 1

D (F)(u,v,w)

Answer 1

Matrix of:

Dx/du dx/dv dx/dw
Dy/du dy/dv dy/dw
Dz/du dz/dv dz/dw

Question 2

Jacobian of F

Answer 2

Determinant of the derivative matrix

Question 3

Chain rule from GoF

Answer 3

D (GoF)(u,v,w) = D (G)(F (u,v,w))D (F)(u,v,w)

Question 4

Coordinate surface

Answer 4

For coordinate systems in R^3, if one variable is fixed, the result is a coordinate surface

Question 5

Polar coordinates

Cylindrical polars

Answer 5

x=rcos (theta)
y=rsin (theta)
r = sqrt (x^2 + y^2) theta=tan^-1 (y/x)
Jacobian = r

x and y stay the same, z = z

Question 6

D (F^-1)(F (u,v))

d (u,v) / d(x,y) * F (u,v

Answer 6

= (D (F)(u,v))^-1

= (d (x,y) / d (u,v)) ^-1