Partial Derivative of Polar Coordinates Proof

**ghow90** · April 7th 2010, 07:16 AM

The question is

Let $z=f(x,y)$ be a differential function of x and y and let x=rcosθ and y=rsinθ

Show that $(\frac{\partial z}{\partial x})^2+(\frac{\partial z}{\partial y})^2=(\frac{\partial z}{\partial r})^2+\frac{1}{r^2}(\frac{\partial z}{\partial \theta})^2$

**Jester** · April 7th 2010, 08:02 AM

Originally Posted by ghow90

The question is

Let $z=f(x,y)$ be a differential function of x and y and let x=rcosθ and y=rsinθ

Show that $(\frac{\partial z}{\partial x})^2+(\frac{\partial z}{\partial y})^2=(\frac{\partial z}{\partial r})^2+\frac{1}{r^2}(\frac{\partial z}{\partial \theta})^2$

Use Jacobians to determine how the derivatives change.

$<br /> z_x = \frac{\partial(z,y) }{\partial(x,y)} = \frac{\partial(z,y)}{\partial(r,\theta)}/ \frac{\partial(x,y)}{\partial(r,\theta)} = \frac{z_r y_{\theta} - z_{\theta} y_r}{x_r y_{\theta} - x_{\theta} y_r} = \cos \theta z_r - \frac{\sin \theta}{r} z_{\theta}.<br />$

Similar for $z_y$ . Then square both, add together and simplify.

Math Help - Partial Derivative of Polar Coordinates Proof

LinkBack

Thread Tools

Display

Partial Derivative of Polar Coordinates Proof

Similar Math Help Forum Discussions

Showing polar coordinate partial derivative stuff

Transforming integral in cartesian coordinates to polar coordinates

Proof of polar coordinates

Epsilon-Delta proof in polar coordinates

polar partial derivative

Search Tags