벡터 미적분 전체 문제

[T] Let $\text{r}:[1,2]\to {\mathbb{R}}^{2}$ be given by $x={e}^{t-1},y=\sin \left(\frac{\pi }{t}\right).$ Use a computer to compute the integral ${\int }_{C}\text{F}\cdot d\text{r}={\int }_{C}2x\,\cos ydx-{x}^{2}\sin ydy,$ where $\text{F}=(2x\,\cos y)\text{i}-({x}^{2}\sin y)\text{j}.$

[T] Use a computer algebra system to find the mass of a wire that lies along curve $\text{r}(t)=({t}^{2}-1)\text{j}+2t\text{k},0\le t\le 1,$ if the density is $\frac{3}{2}t.$

Find the circulation and flux of field $\text{F}=-y\text{i}+x\text{j}$ around and across the closed semicircular path that consists of semicircular arch ${\text{r}}_{1}(t)=(a\cos t)\text{i}+(a\sin t)\text{j},0\le t\le \pi ,$ followed by line segment ${\text{r}}_{2}(t)=t\text{i},-a\le t\le a.$

For the following exercises, show that the following vector fields are conservative by using a computer. Calculate ${\int }_{C}\text{F}\cdot d\text{r}$ for the given curve.

Compute ${\int }_{C}\cos x\,\cos ydx-\sin x\,\sin ydy,$ where $\text{c}(t)=<t,{t}^{2}>,0\le t\le 1.$

Complete the proof of (see text) by showing that ${f}_{y}=Q(x,y).$

${\int }_{C}^{}2xydx+(x+y)dy,$ where C is the path from (0, 0) to (1, 1) along the graph of $y={x}^{3}$ and from (1, 1) to (0, 0) along the graph of $y=x$ oriented in the counterclockwise direction

${\int }_{C}^{}2xydx+(x+y)dy,$ where C is the boundary of the region lying between the graphs of $y=0$ and $y=4-{x}^{2}$ oriented in the counterclockwise direction

${\int }_{C}^{}2\,\arctan \left(\frac{y}{x}\right)dx+\ln ({x}^{2}+{y}^{2})dy,$ where C is defined by $x=4+2\,\cos \,\theta ,y=4\,\sin \,\theta$ oriented in the counterclockwise direction

${\int }_{C}^{}\sin \,x\,\cos \,ydx+(xy+\cos \,x\,\sin \,y)dy,$ where C is the boundary of the region lying between the graphs of $y=x$ and $y=\sqrt{x}$ oriented in the counterclockwise direction

${\int }_{C}^{}xydx+(x+y)dy,$ where C is the boundary of the region lying between the graphs of ${x}^{2}+{y}^{2}=1$ and ${x}^{2}+{y}^{2}=9$ oriented in the counterclockwise direction

${\int }_{C}(-ydx+xdy),$ where C consists of line segment C₁ from $(-1,0)$ to (1, 0), followed by the semicircular arc C₂ from (1, 0) back to (–1, 0)

Let C be the curve consisting of line segments from (0, 0) to (1, 1) to (0, 1) and back to (0, 0). Find the value of ${\int }_{C}xydx+\sqrt{{y}^{2}+1}dy.$

Evaluate line integral ${\int }_{C}x{e}^{-2x}dx+({x}^{4}+2{x}^{2}{y}^{2})dy,$ where C is the boundary of the region between circles ${x}^{2}+{y}^{2}=1$ and ${x}^{2}+{y}^{2}=4,$ and is a positively oriented curve.

Find the counterclockwise circulation of field $\text{F}(x,y)=xy\text{i}+{y}^{2}\text{j}$ around and over the boundary of the region enclosed by curves $y={x}^{2}$ and $y=x$ in the first quadrant and oriented in the counterclockwise direction.

Evaluate ${\int }_{C}{y}^{3}dx-{x}^{3}{y}^{2}dy,$ where C is the positively oriented circle of radius 2 centered at the origin.

Evaluate ${\int }_{C}{y}^{3}dx-{x}^{3}dy,$ where C includes the two circles of radius 2 and radius 1 centered at the origin, both with positive orientation.

Calculate ${\int }_{C}-{x}^{2}ydx+x{y}^{2}dy,$ where C is a circle of radius 2 centered at the origin and oriented in the counterclockwise direction.

Calculate integral ${\int }_{C}2[y+x\,\sin (y)]dx+[{x}^{2}\cos (y)-3{y}^{2}]dy$ along triangle C with vertices (0, 0), (1, 0) and (1, 1), oriented counterclockwise, using Green’s theorem.

Evaluate integral ${\int }_{C}({x}^{2}+{y}^{2})dx+2xydy,$ where C is the curve that follows parabola $y={x}^{2}\,\text{from}\,(0,0)\text{ to }(2,4),$ then the line from (2, 4) to (2, 0), and finally the line from (2, 0) to (0, 0).

Evaluate line integral ${\int }_{C}(y-\sin (y)\cos (y))dx+2x\,{\sin }^{2}(y)dy,$ where C is oriented in a counterclockwise path around the region bounded by $x=-1,x=2,y=4-{x}^{2},$ and $y=x-2.$