91影视

Evaluate$\mathbf{}\mathbf{鈭�}\mathit{V}\mathbf{鈰�}\mathit{n}\mathit{d}\mathit{蟽}$over the curved surface of the hemisphere${\mathit{x}}^{\mathbf{2}}\mathbf{+}{\mathit{y}}^{\mathbf{2}}\mathbf{+}{\mathit{z}}^{\mathbf{2}}\mathbf{=}\mathbf{9}\mathbf{,}\mathit{z}\mathbf{鈮�}\mathbf{0}$, if$\mathbf{V}\mathbf{=}\mathbf{yi}\mathbf{+}\mathbf{xzj}\mathbf{+}\mathbf{(}\mathbf{2}\mathbf{z}\mathbf{}\mathbf{-}\mathbf{}\mathbf{1}\mathbf{)}\mathbf{k}$.Careful: See Problem 9.

(a) Given $\mathbf{桅}\mathbf{=}{\mathbf{x}}^{\mathbf{2}}\mathbf{-}{\mathbf{y}}^{\mathbf{2}}$, sketch on one graph the curves$\mathbf{桅}\mathbf{=}\mathbf{4}\mathbf{,}\mathbf{桅}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{桅}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{桅}\mathbf{=}\mathbf{-}\mathbf{1}\mathbf{,}\mathbf{桅}\mathbf{=}\mathbf{-}\mathbf{4}$. If$\mathbf{桅}$is the electrostatic potential, the curves$\mathbf{桅}\mathbf{=}$const. are equipotential, and the electric field is given by$\mathbf{E}\mathbf{=}\mathbf{-}\mathbf{鈭�}\mathbf{桅}$. If$\mathbf{桅}$is temperature, the curves$\mathbf{桅}$= const. are isothermals and$\mathbf{鈭�}\mathbf{桅}$is the temperature gradient; heat flows in the direction$\mathbf{-}\mathbf{鈭�}\mathbf{桅}$.

(b) Find and draw on your sketch the vectors$\mathbf{-}\mathbf{鈭�}\mathit{桅}$at the points$\left(x,y\right)\mathbf{=}\left(\underline{+}1,\underline{+1}\right)$,$\left(0,\underline{+}2\right)\mathbf{,}\left(\underline{+}2,0\right)$,. Then, remembering that$\mathbf{鈭�}\mathbf{桅}$is perpendicular to$\mathbf{桅}$= const., sketch, without computation, several curves along which heat would flow [see (a)].

Verify that the force field is conservative. Then find a scalar potential 蠁 such that$\mathbf{F}\mathbf{=}\mathbf{-}\mathbf{鈭�}\mathbf{蠁}\mathbf{,}\mathbf{F}\mathbf{=}\mathbf{yi}\mathbf{+}\mathbf{xj}\mathbf{+}\mathbf{k}$ ,

K = constant.

A cylindrical capacitor consists of two long concentric metal cylinders. If there is a charge of k coulombs per meter on the inside cylinder of radius, ${\mathit{R}}_{\mathbf{1}}$and coulombs per meter on the outside cylinder of radius,${\mathit{R}}_{\mathbf{1}}$find -k the electric field E between the cylinders. Hint: Use Gauss鈥檚 law and the method indicated in Figure 10.7. What is E inside the inner cylinder? Outside the outer cylinder? (Again use Gauss鈥檚 law.) Find, either by inspection or by direct integration, the potential role="math" localid="1659237306724" $\mathit{蠁}$such that$\mathit{E}\mathbf{}\mathbf{=}\mathbf{}\mathbf{-}\mathbf{鈭�}\mathit{蠁}$for each of the three regions above. In each case E is not affected by adding an arbitrary constant to$\mathit{蠁}$. Adjust the additive constant to make$\mathit{蠁}$a continuous function for all space

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

Given ${\mathbf{F}}_{\mathbf{1}}=\mathbf{2}\mathbf{xi}-\mathbf{2}\mathbf{yzj}-{\mathbf{y}}^{\mathbf{2}}\mathbf{k}$and${\mathit{F}}_{\mathbf{2}}\mathbf{=}\mathit{y}\mathit{i}\mathbf{-}\mathit{x}\mathit{i}$

(a) Are these forces conservative? Find the potential corresponding to any conservative force.

(b) For any nonconservative force, find the work done if it acts on an object moving from (-1,-1)to (1,1)along each of the paths shown.

Expand the triple product for $\mathbf{a}\mathbf{=}\mathbf{蝇}\mathbf{脳}\left(蝇脳r\right)$ given in the discussion of Figure 3.8. If is perpendicular to $\mathbf{w}$(Problem 16), show that $\mathbf{a}\mathbf{=}\mathbf{鈥�}{\mathbf{蝇}}^{\mathbf{2}}\mathbf{r}$, and so find the elementary result that the acceleration is toward the center of the circle and of magnitude $\raisebox{1ex}{${\mathbf{v}}^{\mathbf{2}}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{r}$}\right.$.

For the force field $\mathit{F}\mathbf{=}\mathbf{-}\mathit{y}\mathit{i}\mathbf{+}\mathit{x}\mathit{j}\mathbf{+}\mathit{z}\mathit{k}\mathbf{,}$ calculate the work done in moving a particle from (1,0,0) torole="math" localid="1664273455603" $\left(-1,0,蟺\right)$

(a) along the helix$\mathit{x}\mathbf{=}\mathit{c}\mathit{o}\mathit{s}\mathit{t}\mathbf{,}\mathit{y}\mathbf{=}\mathit{s}\mathit{i}\mathit{n}\mathit{t}\mathbf{,}\mathit{z}\mathbf{=}\mathit{t}\mathbf{;}$

(b) along the straight line joining the points.

If A and B are the diagonals of a parallelogram, find a vector formula for the area of the parallelogram.

Given, integrate $\mathit{V}\mathbf{}\mathbf{-}\mathit{n}\mathit{d}\mathbf{}\mathit{蟽}$over the whole surface of the cube of side 1 with four of its vertices at $\left(0,0,0\right)\mathbf{,}\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{)}\mathbf{,}\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{0}\mathbf{)}\mathbf{,}\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{)}\mathbf{,}$Evaluate the same integral by means of the divergence theorem.