Chapter 4

Let \(\mathrm{u}=\mathrm{u}(\mathrm{x}, \mathrm{y})\) be implicitly defined as a function of \(\mathrm{x}\) and \(\mathrm{y}\) by the equation \(u+\ln u \approx x y\). Find \((\partial \mathrm{u} / \partial \mathrm{x}),(\partial \mathrm{u} / \partial \mathrm{y}),\left[\left(\partial^{2} \mathrm{u}\right) /(\partial \mathrm{x} \partial \mathrm{y})\right]\) and \(\left[\left(\partial^{2} \mathrm{u}\right) /(\partial \mathrm{y} \partial \mathrm{x})\right]\)

1 1 ] dill [ 2 Let \(\mathrm{u}(\mathrm{x}, \mathrm{y})\) and \(\mathrm{v}(\mathrm{x}, \mathrm{y})\) be defined as functions of \(\mathrm{x}\) and \(\mathrm{y}\) by the equations \(u \cos v-x=0\) (1) \(u \sin v-y=0\) (2) Find \((\partial \mathrm{u} / \partial \mathrm{x}) \quad\) and \(\quad(\partial \mathrm{v} / \partial \mathrm{x})\)

Define the partial derivative of a real valued function, \(\mathrm{f}\). Then compute \((\partial \mathrm{f} / \partial \mathrm{x})\) and \((\partial \mathrm{f} / \partial \mathrm{y})\) where , b) \(\mathrm{f}(\mathrm{x}, \mathrm{y})=\cos [\mathrm{x}(1+\mathrm{y})]\), c) \(\mathrm{f}(\mathrm{x}, \mathrm{y})=\mathrm{x}^{\mathrm{y}}\) a) \(f(x, y)=x y\)

Find the differential of the vector function \(\mathrm{f} \rightarrow \mathrm{R}^{3} \rightarrow \mathrm{R}^{3}\) defined by Evaluate \(\mathrm{f}(2,1,1)\) and then find \(\mathrm{f}^{\rightarrow}(2.01,1.03,1.02)\) exactly and approximately using differentials.

Find the Jacobian matrix and the differential of the vector Function \(\mathrm{f}: \mathrm{R}^{2} \rightarrow \mathrm{R}^{3}\) defined by \(|\mathrm{x}(\mathrm{u}, \mathrm{v})| \quad|\cos \mathrm{u} \cos \mathrm{v}|\) \(\mathrm{f}(\mathrm{u}, \mathrm{v})=|\mathrm{y}(\mathrm{u}, \mathrm{v})|=|\cos \mathrm{u} \sin \mathrm{v}|\) \(|\mathbf{z}(\mathrm{u}, \mathrm{v})| \quad \sin \mathrm{u}\)

Compute the Jacobian determinant of the transformation from a) cylindrical coordinates to Cartesian coordinates, b) spherical coordinates to Cartesian coordinates.

Let \(\mathrm{F}(\mathrm{x}, \mathrm{y})\) be differentiable in \(\mathrm{x}\) and \(\mathrm{y}\) and introduce polar coordinates \(\mathrm{r}, \theta\) by writing \(\mathrm{x}=\mathrm{r} \cos \theta, \mathrm{y}=\mathrm{r} \sin \theta\). Find \(\partial \mathrm{F} / \partial \mathrm{r}\) and \(\partial \mathrm{F} / \partial \theta\) in terms of \(\partial \mathrm{F} / \partial \mathrm{x}\) and \(\partial \mathrm{F} / \partial \mathrm{y}\).

Let \(\begin{aligned} \mathrm{Y}^{-}\left(\mathrm{U}^{-}\right) &=\left|\mathrm{y}_{1}\left(\mathrm{u}_{1}, \mathrm{u}_{2}, \mathrm{u}_{3}\right)\right| \\\\\left|\mathrm{y}_{2}\left(\mathrm{u}_{1}, \mathrm{u}_{2}, \mathrm{u}_{3}\right)\right| &\left|\mathrm{u}_{1} \mathrm{u}_{3}+\mathrm{u}^{2}{\underline{\phantom{xx}}}_{2}\right| \end{aligned}\) and \(\mathrm{U}^{-}\left(\mathrm{X}^{-}\right)=\left|\mathrm{u}_{1}\left(\mathrm{x}_{1}, \mathrm{x}_{2}\right)\right|=\left|\mathrm{x}_{1} \cos \mathrm{x}_{2}+\left(\mathrm{x}_{1}-\mathrm{x}_{2}\right)^{2}\right|\) \(\begin{array}{ll}\left|u_{2}\left(x_{1}, x_{2}\right)\right| & \mid x_{1} \sin x_{2}+x_{1} x_{2} \\ \left|u_{3}\left(x_{1}, x_{2}\right)\right| & \mid x^{2}{\underline{\phantom{xx}}}_{1}-x_{1} x_{2}+x^{2} 2\end{array}\) Find \(\left(\partial \mathrm{y}_{1} / \partial \mathrm{x}_{1}\right)_{(1,0)}=\left.\left(\partial \mathrm{y}_{1} / \partial \mathrm{x}_{2}\right)\right|_{(1,0)},\left.\left(\partial \mathrm{y}_{2} / \partial \mathrm{x}_{1}\right)\right|_{(1,0)}=\left.\left(\partial \mathrm{y}_{2} / \partial \mathrm{x}_{2}\right)\right|_{(1,0)}\)

Let \(F=F(x, y)\) have second partial derivatives in a region of the \(\mathrm{xy}\) -plane and introduce polar coordinates in this region by writing \(\mathrm{x}=\mathrm{r} \cos \theta, \mathrm{y}=\mathrm{r} \sin \theta\). Find \(\partial^{2} \mathrm{~F} / \partial \mathrm{r} \partial \theta\) in terms of derivatives of \(\mathrm{F}\) with respect to \(\mathrm{x}\) and \(\mathrm{y}\).

Use differentials to compute a) \((\partial z / \partial \mathrm{x}),(\partial z / \partial y)\) where \(z=\left[\left(x^{2}-1\right) / y\right]\) b) \((\partial r / \partial x),(\partial r / \partial y),(\partial x / \partial r)\) where \(r=\sqrt{\left(x^{2}+y^{2}\right)}\) c) \((\partial \mathrm{z} / \partial \mathrm{x}),(\partial \mathrm{z} / \partial \mathrm{y})\) where \(\mathrm{z}=\arctan (\mathrm{y} / \mathrm{x})\)

Let \(\mathrm{u}=\mathrm{F}(\mathrm{x}, \mathrm{y})\) have second partial derivatives in a region of the \(\mathrm{xy}\) -plane and introduce the change of variables $$ \mathrm{x}=\mathrm{s}+\mathrm{t}, \quad \mathrm{y}=\mathrm{s}-\mathrm{t} $$ in that region so that \(\mathrm{u}=\mathrm{F}(\mathrm{x}(\mathrm{s}, \mathrm{t}), \mathrm{y}(\mathrm{s}, \mathrm{t}))\). Find $$ \left(\partial^{2} \mathrm{u} / \partial \mathrm{x}^{2}\right)-\left(\partial^{2} \mathrm{u} / \partial \mathrm{y}^{2}\right) $$ in terms of derivatives with respect to \(\mathrm{s}\) and \(\mathrm{t}\).

If \(\mathrm{G}(\mathrm{s}, \mathrm{t})=\mathrm{F}\left(\mathrm{e}^{\mathrm{s}} \cos \mathrm{t}, \mathrm{e}^{\mathrm{s}} \sin \mathrm{t}\right)\), show that \(\mathrm{G}_{11}+\mathrm{G}_{22}=\mathrm{e}^{2 \mathrm{~s}}\left(\mathrm{~F}_{11}+\mathrm{F}_{22}\right)\), where \(\mathrm{G}_{11}\) and \(\mathrm{G}_{22}\) are evaluated at \((\mathrm{s}, \mathrm{t})\) and \(\mathrm{F}_{11}\) and \(\mathrm{F}_{22}\) are evaluated at \(\left(\mathrm{e}^{3} \cos \mathrm{t}, \mathrm{e}^{\mathrm{s}} \sin \mathrm{t}\right)\). Note: \(\mathrm{G}_{11}, \mathrm{G}_{22}\) are abbreviations for \(\left(\partial^{2} \mathrm{G} / \partial \mathrm{s}^{2}\right),\left(\partial^{2} \mathrm{G} / \partial \mathrm{t}^{2}\right)\) respectively and \(F_{11}, F_{22}\) are abbreviations for \(\left(\partial^{2} F / \partial s^{2}\right)\) and \(\left(\partial^{2} \mathrm{~F} / \partial \mathrm{t}^{2}\right)\), respectively.

Show that if a function \(\mathrm{F}: \mathrm{R}^{2} \rightarrow \mathrm{R}\) satisfies \(\partial \mathrm{F} / \partial \mathrm{x}=\partial \mathrm{F} / \partial \mathrm{y}\), then \(F(x, y)=f(x+y)\) where \(f\) is some differentiable function of one variable, \(\mathrm{s}=\mathrm{x}+\mathrm{y}\)

Show that the line normal to the surface given by \(\mathrm{F}(\mathrm{x}, \mathrm{y}, \mathrm{z})=0\) at a point \(\left(\mathrm{x}_{0}, \mathrm{y}_{0}, \mathrm{z}_{0}\right)\) has direction ratios \((\partial \mathrm{F} / \partial \mathrm{x})|[(\mathrm{x}) 0,(\mathrm{y}) 0,(z) 0]:(\partial \mathrm{F} / \partial \mathrm{y})|[(\mathrm{x}) 0,(\mathrm{y}) 0,(\mathrm{z}) 0]:(\partial \mathrm{F} / \partial \mathrm{z}) \mid[(\mathrm{x}) 0,(\mathrm{y}) 0,(\mathrm{z}) 0]\) What are these ratios when \(z=f(x, y)\) is a solution to \(\mathrm{F}(\mathrm{x}, \mathrm{y}, \mathrm{z})=0 ?\)

Find the equation of the tangent plane to the surface described by the equation \(\mathrm{F}(\mathrm{x}, \mathrm{y}, \mathrm{z})=0\) at the point \(\left(\mathrm{x}_{0}, \mathrm{y}_{0}, \mathrm{z}_{0}\right)\). What is this equation if \(\mathrm{z}=\mathrm{f}(\mathrm{x}, \mathrm{y})\) is a solution to \(\mathrm{F}(\mathrm{x}, \mathrm{y}, \mathrm{z})=0 ?\)

Find the line orthogonal to the graph of \(\mathrm{f}(\mathrm{x}, \mathrm{y})=\mathrm{xy}\) at the point \(\left(\mathrm{x}_{0}, \mathrm{y}_{0}, \mathrm{z}_{0}\right)=(-2,3,-6)\).

Find the equation of the tangent plane to the graph of $$ z=f(x, y)=x^{2}+2 y^{2}-1 $$ at the points (a) \(\left(\mathrm{x}_{0}, \mathrm{y}_{0}\right)=(0,0)\) (b) \(\left(\mathrm{x}_{0}, \mathrm{y}_{0}\right)=(1,1)\).

(a) Let \(\mathrm{f}(\mathrm{x}, \mathrm{y})=\mathrm{r}=\sqrt{\left(\mathrm{x}^{2}+\mathrm{y}^{2}\right)}\). Find grad \(\mathrm{f}\left(\mathrm{X}^{-}\right)\) where \(\mathrm{X}^{-}=(\mathrm{x}, \mathrm{y})\) (b) Let \(\mathrm{f}(\mathrm{x}, \mathrm{y})=\sin \mathrm{r}=\sin \sqrt{\left(\mathrm{x}^{2}+\mathrm{y}^{2}\right)}\). Find grad \(\mathrm{f}\left(\mathrm{X}^{-}\right)\). (c) State a general theorem which encompasses the results of (a) and (b).

Let \(u=r^{3}\) be a scalar field where \(r\) is the distance \(O P\) from the origin 0 to a variable point \(\mathrm{P}\), in \(\mathrm{R}^{3}\). Find the gradient of \(u\) at \(P\) without resorting to rectangular coordinates.

Define the term directional derivative and use the definition to compute the derivative of \(\mathrm{f}(\mathrm{x}, \mathrm{y})=\mathrm{x}^{2}+3 \mathrm{xy}\) in the direction \(\beta^{-}=[(1 / \sqrt{2}),-(1 / \sqrt{2})]\) at the point \(p \overrightarrow{0}=(2,0)\)

Derive the formula \(\mathrm{D}_{\mathrm{V}} \rightarrow \mathrm{F}\left(\mathrm{X}^{-}\right)=\nabla^{-} \mathrm{F} \cdot \mathrm{V}^{-}=\) grad \(\mathrm{F} \cdot \mathrm{V}^{-}\) where \(F\) is a real valued function of \(n\) variables, \(F: R^{n} \rightarrow R\); \(\mathrm{V}^{-}\) is a unit vector in \(\mathrm{R}^{\mathrm{n}}\), i.e., \(\left|\mathrm{V}^{-}\right|=1 ; \mathrm{X}^{-}=\left(\mathrm{x}_{1}, \mathrm{x}_{2}, \ldots, \mathrm{x}_{\mathrm{n}}\right)\) is the directed vector of the point \(\left(\mathrm{x}_{1}, \ldots, \mathrm{x}_{\mathrm{n}}\right)\) and \(\mathrm{D}_{\mathrm{V}} \rightarrow \mathrm{F}\left(\mathrm{X}^{-}\right)\) is the directional derivative of \(\mathrm{F}\) at \(\mathrm{X}^{-}\) in the direction \(\mathrm{V}^{-}\). Then use formula (1) to find the directional derivative of \(F\left(X^{-}\right)=F(x, y)=x^{2}+y^{3}\) at \((-1,3)\) in the direction of \(\mathrm{V}^{-}=(1,2)\)

Compute the directional derivative of \(\mathrm{f}(\mathrm{x}, \mathrm{y})=\mathrm{x}^{2}+\mathrm{y}^{2}\) at the point \((-1,3)\) in the direction of maximal increase of \(f\) Explain your reasoning.

Give a definition of a potential function, \(\varphi\), of a vector field \(\mathrm{F}^{-}\left(\mathrm{X}^{-}\right)\). Determine whether or not \(\mathrm{F}^{-}(\mathrm{x}, \mathrm{y})=\left(\mathrm{e}^{\mathrm{xy}}, \mathrm{e}^{\mathrm{x}+\mathrm{y}}\right)\) has a potential function.

Determine whether the following vector fields have potential functions and if so find them: a) \(F(x, y)=\left(3 x y, \sin x y^{3}\right)\) b) \(F(x, y)=\left(2 x y, x^{2}+3 y^{2}\right)\)

Find a potential function for the vector field \(F^{-}(x, y, z)=\left(y \cos (x y), x \cos (x y)+2 y z^{3}, 3 y^{2} z^{2}\right)\)

Determine whether or not the vector field \(F(x, y)=\left[\left\\{\left(e^{T} x\right) / r\right\\},\left\\{\left(e^{\mathrm{r}} y\right) / r\right\\}\right], r=\sqrt{\left(x^{2}+y^{2}\right)}\) has a potential function (i.e., if it is conservative). If a potential function exists, find it.