In changing from Cartesian to polar coordinates, $\frac { \partial u } { \partial y }$ is
A) $\cos \theta \frac { \partial u } { \partial r } - \sin \theta \frac { \partial u } { \partial \theta } / r$
B) $\cos \theta \frac { \partial u } { \partial r } + \cos \theta \frac { \partial u } { \partial \theta } / r$
C) $\sin \theta \frac { \partial u } { \partial r } + \cos \theta \frac { \partial u } { \partial \theta }$
D) $\sin \theta \frac { \partial u } { \partial r } + \cos \theta \frac { \partial u } { \partial \theta } / r$
E) $\sin \theta \frac { \partial u } { \partial r } - \cos \theta \frac { \partial u } { \partial \theta } / r$

Question

Quizplus · Accepted Answer

In polar coordinates, the transformation of partial derivatives involves both the radial and angular components. The correct transformation for $\frac{\partial u}{\partial y}$ in terms of polar coordinates is $\sin \theta \frac{\partial u}{\partial r} + \cos \theta \frac{\partial u}{\partial \theta} / r$, which accounts for the change in the y-direction due to both a change in radius and a change in angle, properly scaled by the radius $r$ for the angular component.

In Changing from Cartesian to Polar Coordinates ∂u∂y\frac { \partial u } { \partial y }∂y∂u​

In Changing from Cartesian to Polar Coordinates $\frac { \partial u } { \partial y }$