Solution to "Find Jacobian for Cartesian to Cylindrical Polar Coordinates

[adichy]

Homework Statement

Consider the scalar field Φ(x, y, z) given by
Φ(x, y, z) = x^2y^2z^2
(a) Write down the vector field A(x, y, z) defined by
A(x, y, z) = ∇Φ(x, y, z)
(b) Write down the scalar field Q(x, y, z) defined by
Q(x, y, z) = ∇ . A(x, y, z)
(c) Find the Jacobian for the transformation from Cartesian coordinates (x, y, z) to
cylindrical polar coordinates (ρ, φ, z), where
x = ρ cos φ
y = ρ sin φ
z = z

Homework Equations

The Attempt at a Solution

ive done part a and b might have got it

for a) A(x,y,z)=2xy^2z^2 i + 2yx^2z^2 j + 2zx^2y^2
b) Q(x,y,z)=2y^2z^2 i + 2x^2z^2 j + 2x^2y^2
c) i kno to find the jacobian i need to partiali differentiate the cartesian and partialli differentiate the cylindrical polar and then put it in a matrix and then find the determinant, problem is i dnt kno wat to differentiate as in which one is the cartesian and the 9 variables to put in the matrix

 

[HallsofIvy]

The Jacobian is
$$\begin{bmatrix}\frac{\partial x}{\partial r} & \frac{\partial x}{\partial \theta} & 0 \\ \frac{\partial y}{\partial r} & \frac{\partial y}{\partial \theta} & 0 \\ 0 & 0 & 1 \end{bmatrix}$$

 

[adichy]

which variables are the ones that differentiated:

with respect to r
with respect to \\theta (from the variables i was given i didnt c ne \\thetas :S)

thx

 

[HallsofIvy]

I just noticed that I had thought you were using spherical coordinates when you are actually using polar coordinates. I have now editted it. But even though my first post was not exactly what you wanted, It showed exactly what variables are to be differentiated. Why are you still asking?

 

[adichy]

ahh srry I've jus realized my mistake, i was meant to replace xyz with the given polar coordinates but instead i was getting confused with whether I am meant to put in the vector field i found or the scalar field

thx for ur help