91影视

Let x. Let$\mathit{桅}$represent either temperature or electrostatic potential. Refer to Problem 11 for definitions and find:

(a) The direction in which the temperature is increasing most rapidly at$\mathbf{(}\mathbf{1}\mathbf{,}\frac{\mathbf{-}\mathbf{蟺}}{\mathbf{4}}\mathbf{)}$

and the magnitude of the rate of increase.

(b) The rate of change of temperature with distance at$\mathbf{(}\mathbf{0}\mathbf{,}\frac{\mathbf{蟺}}{\mathbf{3}}\mathbf{)}$in the direction$\mathit{i}\mathbf{+}\mathit{j}\sqrt{\mathbf{3}}$.

(c) The direction and magnitude of the electric field at$\left(0,蟺\right)$.

(d) The magnitude of the electric field at$\mathit{x}\mathbf{=}\mathbf{-}\mathbf{1}$, any y.

Draw a figure similar to figurebut with q outside the surface. A vector (like rin the figure) fromq to the surface now intersects it twice, and for each solid angle $\mathit{d}\mathit{惟}$there are two $\mathit{d}\mathit{惟}$where renters and where it leaves the surface. Show that $\mathit{D}\mathbf{鈰�}\mathit{n}\mathit{d}\mathit{蟽}$is given by (10.21) for the $\mathit{d}\mathit{蟽}$whereleaves r the surface and the negative of(10.21)for the$\mathit{d}\mathit{蟽}$where renters the surface. Hence show that the total$\mathit{d}{\mathit{蟽}}_{\mathbf{2}}$over the closed surface is zero.

Verify that the force field is conservative. Then find a scalar potential $\mathit{胃}$ such that $\mathit{F}\mathbf{=}\mathbf{-}\mathbf{鈭�}\mathit{蠁}\mathbf{,}\mathit{F}\mathbf{=}{\mathit{z}}^{\mathbf{2}}\mathit{s}\mathit{i}\mathit{n}\mathit{h}\mathbf{}\mathit{y}\mathbf{}\mathit{j}\mathbf{+}\mathbf{2}\mathit{z}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathit{h}\mathbf{}\mathit{y}\mathbf{}\mathit{k}\mathbf{,}\mathbf{}\mathit{k}\mathbf{}\mathbf{=}\mathbf{}\mathit{c}\mathit{o}\mathit{n}\mathit{s}\mathit{t}\mathit{a}\mathit{n}\mathit{t}\mathbf{.}$

$\mathbf{鈭�}\mathbf{(}\mathbf{2}\mathbf{xi}\mathbf{鈭�}\mathbf{2}\mathbf{yj}\mathbf{+}\mathbf{5}\mathbf{k}\mathbf{)}\mathbf{脳}\mathbf{苍诲蟽}$over the surface of a sphere of radius $\mathbf{2}$and center at the origin.

If the temperature is $\mathbf{T}\mathbf{=}{\mathbf{x}}^{\mathbf{2}}\mathbf{-}\mathbf{xy}\mathbf{+}{\mathbf{z}}^{\mathbf{2}}$, find

(a) The direction of heat flow at (2,1, -1);

(b) The rate of change of temperature in the direction $\mathbf{j}\mathbf{}\mathbf{-}\mathbf{}\mathbf{k}\mathbf{}\mathbf{at}\mathbf{}\mathbf{(}\mathbf{2}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{-}\mathbf{1}\mathbf{)}$

Verify that the force field is conservative. Then find a scalar potential 蠁 such that

$\mathit{F}\mathbf{=}\mathbf{-}\mathbf{鈭�}\mathit{蠁}\mathbf{,}\mathit{F}\mathbf{=}\frac{\mathbf{y}}{\sqrt{\mathbf{1}\mathbf{-}{\mathbf{x}}^{\mathbf{2}}{\mathbf{y}}^{\mathbf{2}}}}\mathit{i}\mathbf{+}\frac{\mathbf{x}}{\sqrt{\mathbf{1}\mathbf{-}{\mathbf{x}}^{\mathbf{2}}{\mathbf{y}}^{\mathbf{2}}}}\mathit{j}$

Obtain Coulomb鈥檚 law from Gauss鈥檚 law by considering a spherical surface$\mathit{未}$ with centre atq.

$\mathbf{\text{鈭�}}\mathbf{(}\mathbf{yi}\mathbf{鈭�}\mathbf{xj}\mathbf{+}\mathbf{zk}\mathbf{)}\mathbf{鈭�}\mathbf{dr}$around the circumference of the circle of radius $\mathbf{2}$, center at the origin, in the $\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{}\mathbf{y}\mathbf{)}$plane.

(a) Suppose that a hill (as in Fig. 5.1) has the equation $\mathbf{32}\mathbf{-}{\mathbf{x}}^{\mathbf{2}}\mathbf{-}\mathbf{4}{\mathbf{y}}^{\mathbf{2}}$, where $\mathbf{z}\mathbf{=}\mathbf{height}\mathbf{}\mathbf{measured}\mathbf{}\mathbf{from}\mathbf{}\mathbf{some}\mathbf{}\mathbf{refrence}\mathbf{}\mathbf{level}$(in hundreds of feet). Sketch acontour map (that is, draw on one graph a set of curves$\mathbf{z}\mathbf{=}$const.); use the contours $\mathbf{z}\mathbf{=}\mathbf{32}\mathbf{,}\mathbf{19}\mathbf{,}\mathbf{12}\mathbf{,}\mathbf{7}\mathbf{,}\mathbf{0}$(b) If you start at the point$\left(3,2\right)$and in the direction$\mathbf{i}\mathbf{+}\mathbf{j}$, are you going up hillor downhill, and how fast?

Show that$\mathbf{F}\mathbf{=}{\mathbf{y}}^{\mathbf{2}}\mathbf{zsinh}\mathbf{鈦�}\mathbf{\left(}\mathbf{2}\mathbf{xz}\mathbf{\right)}\mathbf{i}\mathbf{+}\mathbf{2}{\mathbf{ycosh}}^{\mathbf{2}}\mathbf{鈦�}\mathbf{\left(}\mathbf{xz}\mathbf{\right)}\mathbf{j}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{xsinh}\mathbf{鈦�}\mathbf{\left(}\mathbf{2}\mathbf{xz}\mathbf{\right)}\mathbf{k}$ is conservative, and find a scalar potential $\mathbf{蠁}$such that $\mathbf{F}\mathbf{=}\mathbf{鈭�}\mathbf{鈭�}\mathbf{蠒}$.