91Ó°ÊÓ

Suppose that the dependencies of the variables can be described informally as " \(w\) depends on all three of \(x, y\), and \(z\) and each in turn depends on both \(s\) and \(t . "\) Express this schematically and write a chain rule for it.

Let \(u\) and \(v\) have continuous derivatives on \(\mathbb{R}\) and let \(f\) be continuous with continuous partial derivatives on \(\mathbb{R}^{2}\). Obtain a formula for \(F^{\prime}(y)\), where $$ F(y)=\int_{u(y)}^{v(y)} f(x, y) d x $$

(Double limits and iterated limits, revisited) A bit of care is needed with the statement "If a double limit exists, so do the two iterated limits, and the two iterated limits equal the double limit." Let $$ f(x, y)=\left\\{\begin{array}{ll} y+x \sin \frac{1}{y} & \text { if } y \neq 0 \\ 0 & \text { if } y=0 \end{array}\right. $$ (a) Show that \(\lim _{(x, y) \rightarrow(0,0)} f(x, y)=0\) but \(\lim _{x \rightarrow 0} \lim _{y \rightarrow 0} f(x, y)\) does not exist.