# CLEAR Calculus

## Lab 5: Just what is the "x-direction" anyway?

Let $$f:\mathbb{R}^2\to\mathbb{R}$$ be a function on the plane. If we let $$x$$ and $$y$$ be the standard cartesian coordinates on the plane, suppose this particular function is given by $f(x,y)={xy^2\over x^2+y^2}{\atop.}$ We may want to compute partial derivatives to help determine how $$f$$ is changing in various directions. Recall that if we compute $$f_x$$ then we get the rate of change in $$f$$ with respect to $$x$$ while $$y$$ remains constant. In all of the problems below $$P$$ is the point given by $$x=1$$ and $$y=1$$ in cartesian coordinates.

1. Compute $$f_x(x,y)$$ and $$f_y(x,y)$$. Evaluate these at the point $$P$$. Draw a figure such as the one to the right and sketch the direction in which $$f_y(x,y)$$ gives the rate of change. The direction for $$f_x(x,y)$$ has already been sketched.

2. There are many other ways in which we can describe this function. For example, we can let $$r$$ and $$\theta$$ be polar coordinates on the plane. Find the polar coordinates for $$P$$. Then using $$x=r\cos\theta$$ and $$y=r\sin\theta$$, rewrite $$f$$ as a function $$f(r,\theta)$$. Compute both partials and evaluate them at $$P$$. On a new figure, sketch and label the directions in which these partials measure the rate of change at $$P$$.

3. Now find $$P$$ in $$x,\theta$$-coordinates. Then use $$x=r\cos\theta$$ to rewrite the polar-coordinate formula for $$f$$ as a function $$f(x,\theta)$$. That is, there can be no $$y$$'s or $$r$$'s in your expression. Compute both partials and evaluate them at $$P$$. On a new figure, sketch and label the directions in which these partials measure the rate of change at $$P$$. (Hint: the direction for $$f_x$$ is NOT straight to the right!)

4. Find $$P$$ in $$y,\theta$$-coordinates. Then use $$y=r\sin\theta$$ to rewrite the polar-coordinate formula for $$f$$ as a function $$f(y,\theta)$$. Compute both partials and evaluate them at $$P$$. Sketch and label the directions in which these partials measure the rate of change at $$P$$.

5. Find $$P$$ in $$x,r$$-coordinates. Then use $$y^2=r^2-x^2$$ to rewrite the original Cartesian formula for $$f$$ as a function $$f(x,r)$$. Compute both partials and evaluate them at $$P$$. Sketch and label the directions in which these partials measure the rate of change at $$P$$.

6. Find $$P$$ in $$y,r$$-coordinates. Then use $$x=\sqrt{r^2-y^2}$$ to rewrite the original Cartesian formula for $$f$$ as a function $$f(y,r)$$. Compute both partials and evaluate them at $$P$$. Sketch and label the directions in which these partials measure the rate of change at $$P$$.

7. Of the twelve directions you sketched at the point $$P$$, there are really only 6 unique directions. Group the twelve partial derivatives by the direction in which they measure rate of change. For example, $$f_y(x,y),$$ $$f_\theta(x,\theta),$$ and $$f_r(x,r)$$ all measure the rate of change in the same direction at $$P$$.

8. Within each of your groupings for question 7 notice that the partials at $$P$$ may be different even though they are supposed to measure the rate of change in the same direction! Explain the difference.