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.
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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. -
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 \(P\). -
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!)
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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 \(P\).
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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 \(P\).
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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 \(P\).
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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 \(P\).
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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.