15.1 Assess Your Understanding

Concepts and Vocabulary

Question

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

Question

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 \(F(x,y,z)=P(x,y,z)+Q(x,y,z)+R(x,y,z)\).

Question

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

Question

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

Skill Building

In Problems 5–14, describe each vector field by drawing some of its vectors.

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

In Problems 15 and 16, use graphing technology to represent each vector field. Then describe the vector field.

Question

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

Question

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

Applications and Extensions

In Problems 17–20, find the gradient vector field of each function \(f.\)

Question

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

Question

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

Question

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

Question

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

Question

  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.

Question

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 \(m\).