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A(n) _____ _______ is a function that associates a vector to a point in the plane or to a point in space.

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True or False Let \(P=P(x,y,z)\), \(Q=Q(x,y,z)\), and \(R=R(x,y,z)\) be functions of three variables defined on a subset \(E\) of space. A vector field over \(E\) is defined as the function \(F(x,y,z)=P(x,y,z)+Q(x,y,z)+R(x,y,z)\).

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The domain of the vector field \(\mathbf{F}=\mathbf{F}(x,y)\) is a set of points \((x,y) \) in the plane, and the range of \(\mathbf{F}\) is a set of ______ in the plane.

977

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True or False The gradient of a function \(f\) is an example of a vector field.

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

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

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

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

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

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\(\mathbf{F}=\mathbf{F}(x,y)=-\mathbf{j}\)

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

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

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\(\mathbf{F}=\mathbf{F}(x,y,z)=z \mathbf{k}\)

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\(\mathbf{F}=\mathbf{F}(x,y,z)=x\mathbf{i}\)

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

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\(\mathbf{F}=\mathbf{F}(x,y,z)=-\dfrac{x\mathbf{i}+y\mathbf{j}+z\mathbf{k}}{\sqrt{x^{2}+y^{2}+z^{2}}}\)

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\(f(x,y) =x\sin y+\cos y\)

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\(f(x,y) =xe^{y}\)

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\(f( x,y,z) =x^{2}y+xy+y^{2}z\)

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\(f(x,y,z) =x^{2}y+xyz^{2}\)

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1. Show that the vector field from Example 1, \(\mathbf{F}=\mathbf{F}(x,y)=-y\mathbf{i}+x\mathbf{j}\), is a set of vectors tangent to circles centered at the origin.
2. Confirm that the magnitude of each vector equals the radius of the circle.

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Gravitational Potential Energy  The gravitational field \(\bf{g}\) due to a very small object of mass \(m {\rm kg}\) that is \(r\) meters (m) from a large object is given by \(\mathbf{g}=\!-\dfrac{Gm}{r^{3}}\mathbf{r}\), where \(G=6.67\times\! 10^{-11}{\rm N}{\rm m}^{2}\!/\!{\rm kg}^{2}\) and \(\mathbf{r}={x}\mathbf{i}+y\mathbf{j}+z\mathbf{k}\). Show that the gravitational field is a gradient vector field. That is, show that \(\mathbf{g}=- {\boldsymbol\nabla }u,\) where \(u=-\dfrac{Gm}{r}\). The scalar function \(u\) is called the gravitational potential due to the mass \(m\).