Chapter 6

Calculate the Laplacian ${\mathrm{\nabla }}^2$ of each of the following scalar fields.$x^3-3x{y}^2+y^3$

(a) Given $\varphi =x^2-y^2\epsilon$, find $\mathrm{\nabla }\varphi$ at $(1,1,1)$. (b) Find the directional derivative of $\varphi$ at $(1,1,1)$ in the direction $i-2j+k$. (c) Find the equations of the normal line to the surface $x^2-y^{2}z=0$ at $(1,1,1)$,

$\oint (2ydx-3xdy)$ around the square bounded by $x=3,x=5,y=1$ and $y=3$

Calculate the Laplacian ${\mathrm{\nabla }}^2$ of each of the following scalar fields.$\ln (x^2+y^2)$

Suppose that the temperature in the $(x,y)$ plane is given by $T=xy-x$. Sketch a few isothermal curves, corresponding, for instance, to $T=0,1,2,-1,-2$. Find the direction, in which the temperature changes most rapidly with distance from the point $(1,1)$, and the maximum rate of change. Find the directional derivative of $T$ at $(1,1)$ in the direction of the vector $3\mathrm{i}-4\mathrm{j}$. Heat flows in the direction $-\mathrm{\nabla }T$ (perpendicular to the isothermals). Sketch a few curves along which heat would flow.

Verify that each of the following force fields is conservative. Then find, for each, a scalar potential $\varphi$ such that $\mathbf{F}=-\mathrm{\nabla }\varphi$.$\mathrm{F}=y\sin 2x\mathbf{i}+{\sin }^{2}x\mathrm{j}$

Suppose that the temperature in the $(x,y)$ plane is given by $T=xy-x$. Sketch a few isothermal curves, corresponding, for instance, to $T=0,1,2,-1,-2$. Find the direction, in which the temperature changes most rapidly with distance from the point $(1,1)$, and the maximum rate of change. Find the directional derivative of $T$ at $(1,1)$ in the direction of the vector $3\mathrm{i}-4\mathrm{j}$. Heat flows in the direction $-\mathrm{\nabla }T$ (perpendicular to the isothermals). Sketch a few curves along which heat would flow.

${\int }_c(x\sin x-y)dx+(x-y^2)dy$, where $C$ is the triangle in the $(x,y)$ plane with vertices $(0,0),(1,1)$, and $(2,0)$

Given $u=xy+yz+z\sin x$, find (a) ${\mathbf{V}}_{\mathrm{H}}$ at $(0,1,2);$ (b) the directional derivative of $u$ at $(0,1,2)$ in the direction of $2\mathbf{i}+2\mathrm{j}-\mathrm{k};$ (c) the equations of the tangent plane and of the normal line to the level surface $u=2$ at $(0,1,2)$ (d) a unit vector in the direction of most rapid increase of $u$ at $(0,1,2)$.

Verify that each of the following force fields is conservative. Then find, for each, a scalar potential $\varphi$ such that $\mathbf{F}=-\mathrm{\nabla }\varphi$.$\mathrm{F}=y\mathbf{i}+x\mathbf{j}+\mathbf{k}$