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True or False If \(\nabla \! \ f=\mathbf{F}\) for some function \(f\) that has continuous first-order partial derivatives, then \(\mathbf{F}\) is a potential function for \(f\).

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Multiple Choice If \(\int_{C_{1}}(P\,dx+Q\,dy)=\int_{C_{2}}(P\,dx+Q\,dy)\) for any two piecewise-smooth curves \(C_{1}\) and \(C_{2}\) with the same end points lying entirely in a region \(R,\) then \(\int_{C}(P\,dx+Q\,dy)\) is said to be [(a) one-to-one, (b) singular, (c) independent of \(P\) and \(Q\), (d) independent of the path.]

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True or False The line integral \(\int_{C} \mathbf{F} \, {\cdot}\, d\mathbf{r}\), where \(C\) is a piecewise smooth curve and \(\mathbf{F}\) is a conservative vector field \(\mathbf{,}\) is independent of the path.

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Multiple Choice Suppose \(\mathbf{F}=\mathbf{F}(x,y)\) is a conservative vector field and \(\nabla\! \ f=\mathbf{F}\). The function \(f\) is called a(n) [(a) gradient function, (b) independent function, (c) potential function, (d) gravity function] for \(\mathbf{F}\).

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A piecewise-smooth curve \(C\) is ____________ if its initial point \((x_{0},y_{0})\) and its terminal point \((x_{1},y_{1})\) are the same.

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If \(\mathbf{F}\) is a conservative vector field on some open region \(R\), and \(C\) is a closed, piecewise-smooth curve that lies entirely in \(R\), then \(\int_{C}\mathbf{F}\,{\cdot}\, d\mathbf{r}=\) ____________.

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True or False An open region \(R\) is connected if there are two points \((x_{1},y_{1})\) and \((x_{2},y_{2})\) in \(R\) that can be joined by a piecewise-smooth curve \(C\) that lies entirely in \(R\).

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True or False The converse of the Fundamental Theorem of Line Integrals is true if the open region \(R\) is connected.

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True or False Let \(\mathbf{r}=\mathbf{r}(t)\), \(a\leq t\leq b\), trace out a piecewise-smooth curve \(C\). Then \(C\) is closed if there is a least one point where the curve intersects itself.

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True or False A parabola is a simple curve.

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True or False A region \(R\) is called simply connected if it is closed.

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Let \(\mathbf{F}=\mathbf{F}(x,y)=P(x,y)\mathbf{i}+Q(x,y)\mathbf{j }\) be a vector field, where the functions \(P\) and \(Q\) are continuous on some simply connected,

open region \(R\). Suppose \(\dfrac{\partial P}{\partial y}\) and \(\dfrac{\partial Q}{\partial x}\) are also continuous on \(R\).

Then \(\mathbf{F}(x,y)=P(x,y)\mathbf{i}+Q(x,y)\mathbf{j}\) is a conservative vector field on \(R\) if and only if ____________.

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Find the line integral \(\int_{C}(e^{x}\cos y\,dx-e^{x}\sin y\,dy)\) for each curve \(C\):

1. \(C\) consists of the line segments from \(\left( 0,\dfrac{\pi }{3} \right)\) to \(\left( 1,\dfrac{\pi }{3}\right)\) and from \(\left( 1,\dfrac{ \pi }{3}\right)\) to \((1,\pi )\).
2. \(C\) consists of the line segment from \(\left( 0,\dfrac{\pi }{3}\right)\) to \((1,\pi )\).
3. \(C\) consists of the line segments from \(\left( 0,\dfrac{\pi }{3} \right)\) to \(\left( 0,\dfrac{\pi }{2}\right)\) and from \(\left( 0,\dfrac{\pi }{2}\right)\) to \((1,\pi )\).

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Find the line integral \(\int_{C}(e^{y}\cos x\,dx+e^{y}\sin x\,dy)\) for each curve \(C\):

1. \(C\) consists of the line segment from \((0,0)\) to \((\pi ,1)\).
2. \(C\) consists of the line segments from \((0,0)\) to \(\left( \dfrac{\pi }{2},\dfrac{1}{2}\right)\) and from \(\left( \dfrac{\pi }{2}, \dfrac{1}{2}\right)\) to \((\pi ,1)\).
3. \(C\) consists of the line segments from \((0,0)\) to \(( 0,1)\) and from \(( 0,1)\) to \((\pi ,1)\).

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\(\int_{C} [ ( 2x-y)\, dx+(2y-x)\, dy]\) from \(( 3,-1)\) to \(( 5,4);\quad f(x,y) =x^{2}-xy+y^{2}\)

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\(\int_{C} [( y^{2}+2x)\, dx+2x y\,dy]\) from \(( 2,2)\) to \(( -3,8);\quad (f(x,y) =xy^{2}+x^{2}\)

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\(\int_{C} ( 3x^{2}y\,dx+x^{3}dy)\) from \((0,-1)\) to \(( 3,4);\quad f(x,y) =x^{3}y\)

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\(\int_{C} \big( 2xe^{x^{2}+y^{2}}dx+2y e^{x^{2}+y^{2}}dy\big)\) from \((0,0)\) to \((1,1); f(x,y) =e^{x^{2}+y^{2}}\)

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\(\int_{C} \left( \dfrac{x}{x^{2}+y^{2}}dx+\dfrac{y}{x^{2}+y^{2}}dy\right)\) from \(( 3,4)\) to \(( 5,12)\); \(f(x,y) =\ln \sqrt{x^{2}+y^{2}}\)

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\(\int_{C} \left( \dfrac{x}{\sqrt{x^{2}+y^{2}}}dx+\dfrac{y}{\sqrt{x^{2}+y^{2}}}dy\right)\) from \(( 0,-4)\) to \(( 3,4)\); \(f(x,y) =\sqrt{x^{2}+y^{2}}\)

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\(\int_{C}( x^{3}dx+y^{3}dy)\) from \((1,1)\) to \((2,4)\)

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\(\int_{c}( 2x^{2}dx-4y\,dy)\) from \((-1,2)\) to \(( 2,3)\)

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\(\int_{C}[ ( x^{2}+y^{2})\, dx+ ( 2xy+\sin y)\, dy]\) from \((0,0)\) to \(( 2,0)\)

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\(\int_{C}[ ( x-\cos y)\, dx+x\sin y\,dy]\) from \(\left( 1,\dfrac{\pi }{2}\right)\) to \(( 2,\pi )\)

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\(\int_{C}( y^{2}e^{x}\,dx+2ye^{x}dy)\) from \((0,0)\) to \((\ln 2,2)\)

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\(\int_{C}\left[ xe^{y}dx+\dfrac{1}{2}( x^{2}-4) e^{y}dy\right]\) from \((0,0)\) to \((2,1)\)

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\(\int_{C}( ye^{x}\,dx+ e^{x}\,dy)\) on a closed curve \(C\)

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\(\int_{C}( y\cos x\,dx+\sin x\,dy)\) on a closed curve \(C\)

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\(\mathbf{F}(x,y) =(5x-2y+1) \,\mathbf{i}+( 4-2x) \,\mathbf{j}\)

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\(\mathbf{F}(x,y) =( 4xy) \,\mathbf{i}+(2x^{2}+y) \,\mathbf{j}\)

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\(\mathbf{F}(x,y) =( \ln y+2x) \,\mathbf{i}+\!\left( \dfrac{x}{y}\right)\, \mathbf{j}\)

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\(\mathbf{F}(x,y) =(ye^{x}) \,\mathbf{i}+e^{x}\,\mathbf{j}\)

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\(\mathbf{F}(x,y)=x^{2}\,\mathbf{i}+y^{2}\mathbf{j}\)

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\(\mathbf{F}(x,y)=xy\mathbf{i}+xy\mathbf{j}\)

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\(\mathbf{F}(x,y)=xe^{y}\,\mathbf{i}+\dfrac{1}{2} x^{2}e^{y}\,\mathbf{j}\)

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\(\mathbf{F}(x,y)=(x^{2}+y^{2})\mathbf{i}+(2xy-\sin y)\mathbf{j}\)

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\(\int_{C}(x\,dx+y\,dy),\) where \(C\) is a piecewise-smooth curve joining the points \((1,3)\) and \((2,5)\)

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\(\int_{C}[2xy\,dx+(x^{2}+1)\,dy],\) where \(C\) is a piecewise-smooth curve joining the points \((1,-4)\) and \((-2,3)\)

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\(\int_{C}[(x^{2}+3y)\,dx+3x\,dy],\) where \(C\) is a piecewise-smooth curve joining the points \((1,2)\) and \((-3,5)\)

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\(\int_{C}[(2x+y+1)\,dx+(x+3y+2)\,dy],\) where \(C\) is a piecewise-smooth curve joining the points \((0,0)\) and \((1,2)\)

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\(\int_{C}[(4x^{3}+20xy^{3}-3y^{4})\,dx+(30x^{2}y^{2}-12xy^{3}+5y^{4})\,dy]\) \(,\) where \(C\) is a piecewise-smooth curve joining the points \((0,0)\) and \((1,1)\)

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\(\int_{C}[(2yx^{-1})\,dx+(\ln x^{2})\,dy],\) where \(C\) is a piecewise-smooth curve in the first quadrant joining the points \((1,1)\) and \((5,5)\)

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\(\int_{C}(3x^{2}y^{2}\,dx+2x^{3}y\,dy)\)

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\(\int_{C}[(2x+y)\,dx+(x-2y)\,dy]\)

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\(\int_{C}[(x+3y)\,dx+3x\,dy]\)

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\(\int_{C}[(2x+y)\,dx+(2y+x)\,dy]\)

1002

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\(\int_{C}[(2xy-y^{2})\,dx+(x^{2}-2xy)\,dy]\)

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\(\int_{C}[y^{2}\,dx+(2yx-e^{y})\,dy]\)

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\(\int_{C}[(x^{2}-x+y^{2})\,dx-(ye^{y}-2xy)\,dy]\)

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\(\int_{C}[(3x^{2}y+xy^{2}+e^{x})\,dx+(x^{3}+x^{2}y+\sin y)\,dy]\)

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\(\int_{C}[(y\cos x-2\sin y)\,dx-(2x\cos y-\sin x)\,dy]\)

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\(\int_{C}[(e^{x}\sin y+2y\sin x)\,dx+(e^{x}\cos y-2\cos x)\,dy]\)

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\(\int_{C}[(2x+y\cos x)\,dx+\sin x\,dy]\)

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\(\int_{C}[ (\cos y-\cos x)\,dx+(e^{y}-x\sin y)\,dy]\)

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Show that \(\int_{C}\left( -\dfrac{y}{x^{2}+y^{2}}\,dx+\dfrac{x}{x^{2}+y^{2}}\,dy\right)\) is independent of the path in the rectangle \(R\) whose vertices are \(\left( \dfrac{1}{a},-a\right)\), \((a,-a)\), \(\left(\dfrac{1}{a},-\dfrac{1}{a}\right)\), and \(\left( a,-\dfrac{1}{a}\right),\) \(a>1\). Find a potential function \(f\) for \(\mathbf{F}\).

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Given the constant vector \(\mathbf{c}\), show that \(\mathbf \nabla (\mathbf{c}\,{\cdot}\, \mathbf{r})=\mathbf{c}\), where \(\mathbf{r}=x \mathbf{i}+y\mathbf{j}+z\mathbf{k}\). Use the result to prove that \(\int_{C} \mathbf{c}\,{\cdot}\, d\mathbf{r}=0\), where \(C\) is any closed piecewise smooth curve in space.

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Suppose that \(\mathbf{F}(x,y)\) is a force field directed toward the origin with magnitude inversely proportional to the square of the distance from the origin. (Such “inverse square law” forces are common in nature. See Examples 4 and 5 in Section 15.1, pp. 975-976.)

1. Show that \(\mathbf{F}\) is a conservative vector field.
2. Find a potential function \(f\) for \(\mathbf{F}\).

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Suppose \(f\) and \(g\) are differentiable functions of one variable. Show that \(\int_{C}[f(x)\,dx+g(y)\,dy]=0\), where \(C\) is any circle in the \(xy\)-plane.

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Let \(f\) and \(g\) have continuous partial derivatives in a plane region \(R\), and let \(C\) be a piecewise-smooth curve in \(R\) going from \(A\) to \(B\). Show that \[ \int_{C}f\,{\nabla }g\,{\cdot}\, d\mathbf{r}=f(B)g(B)-f(A)g(A)-\int_{C}g\, \nabla\! \ f\,{\cdot}\, d\mathbf{r} \]