Partial differentiation question?

[applestrudle]

Homework Statement

z = x^2 +y^2

x = rcosθ

y = rsinθ

find partial z over partial x at constant theta

Homework Equations

z = x^2 +y^2

x = rcosθ

y = rsinθ

The Attempt at a Solution

z = 1 + r^2(sinθ)^2

dz/dx = dz/dr . dr/dx

= 2(sinθ)^2r/cosθ

= 2tanθ^2x


the book says 2x[1+2(tanθ)^2]

 

[tiny-tim]

hi applestrudle!

(have a curly d: ∂ and try using the X² button just above the Reply box )

z = x²+y²

x = rcosθ

y = rsinθ

find ∂z/∂x at constant theta

∂/∂x is ambiguous unless you know what the other variables are

it always means that you differentiate wrt x keeping the other variables constant (that's why you need to know what they are!)

in this case, the question tells you the other variable is θ, so first you need to find z(x,θ), ie to write z as a function of x and θ

(hmm … i don't get the result the book gets )

 

[vanhees71]

Here, I'd write $z$ as function of $x$ and $\theta$. Then it's easy to take the partial derivative. Obviously we have
$$z=r^2=\frac{x^2}{\cos^2 \theta}.$$
Then you can take the partial derivative wrt. $x$ and fixed $\theta$ easily, but what you quoted as solution of the book is obviously wrong (the factor 2 in front of $\tan^2 \theta$ should not be there).