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) \(\left(\mathrm{x}_{0}, \mathrm{y}_{0}\right)=(1,1)\).

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

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 \(\mathrm{F}(\mathrm{x}, \mathrm{y}, \mathrm{z})=0 ?\)

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 \(p \overrightarrow{0}=(2,0)\)

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.

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 \(\mathrm{y}\).