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Chapter 14: Q. 4 (page 1095)

What are the inputs of a vector field in $ℝ^{3}$ ?

Short Answer

Expert verified

The inputs of a vector field in $ℝ^{3}$ is points in $ℝ^{3}$ .

Step by step solution

01

Step 1.Given Information

What are the inputs of a vector field in $ℝ^{3}$ ?

02

Step 2. The input of a vector field

We've looked at functions with two and three variables that have a point as their input (either a point in the plane or in space). We could also create functions with a point as the input (again, either in the plane or in space).

Hence, a vector field in $ℝ^{3}$ has inputs that are points in $ℝ^{3}$ .

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Most popular questions from this chapter

$Use the vector field F (x, y) = x^{2} e^{y} i + \cos (x) \sin (y) j and Green ’ s Theorem to write the line integral of F (x, y) about the unit circle, traversed counterclockwise, as a double integral . Do not evaluate the integral .$

Find the masses of the lamina:

The lamina occupies the region of the hyperboloid with equation $z^{2} + 4 = x^{2} + y^{2}$ that lies above and/or below the disk of radius 5 about the origin in the XY-plane, and the density function, ρ(x, y,z), is proportional to the distance from the origin.

Compute the curl of the vector fields:

$F (x, y) = \cos (x + y) i + \sin (x - y) j$ .

$Compute the curl of the vector fields in Exercises 23 - 28 . F (x, y, z) = \sin^{- 1} (xy) i + \ln (y + z) j + \frac{1}{2 x + 3 y + 5 z + 1} k$

Give a smooth parametrization of the upper half of the unit sphere in terms of x and y.

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