Chapter 6: Q22P (page 335)

Find vector fields $A$ such that $V = curl A$ for each given V. $V = (y + z) i + (x - z) j + (x^{2} + y^{2}) k$

Short Answer

Expert verified

The vector field derived is $A = i (xz - \frac{y^{3}}{3}) + j (- yz + \frac{x^{3}}{3}) + k (x + y) z + \nabla u$ .

Step by step solution

Given Information.

The given equation is $V = (y + z) i + (x - z) j + (x^{2} + y^{2}) k$ .

Definition of vector.

A quantity that has magnitude as well as direction is called a vector. It is typically denoted by an arrow in which the head determines the direction of the vector and the length determines its magnitude.

Find the vector field.

In this problem find vector $A$ , whose curl gives the vector

$V = (y + z) i + (x - z) j + (x^{2} + y^{2}) k$

The components of vector are given in terms of the derivatives of $A$ .

$\frac{\partial A_{z}}{\partial y} - \frac{\partial A_{y}}{\partial z} = y + z$

$\frac{\partial A_{x}}{\partial y} - \frac{\partial A_{z}}{\partial z} = x - z$

$\frac{\partial A_{y}}{\partial x} - \frac{\partial A_{x}}{\partial y} = x^{2} + y^{2}$

Observe the first two equation $A_{z} = Z (x + y)$ , and choose the equation.

The first equation then gives the equation $- \frac{\partial A_{y}}{\partial z} = y$ .

Integrate to obtain the equation $A_{y} = - yz + f (x, y)$ .

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Find vector fields $A$ such that $V = curl A$ for each given V. $V = (y + z) i + (x - z) j + (x^{2} + y^{2}) k$

Short Answer

Step by step solution

Given Information.

Definition of vector.

Find the vector field.

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Find vector fieldsA such that V=curlAfor each given V.V=(y+z)i+(x−z)j+x2+y2k

Short Answer

Step by step solution

Given Information.

Definition of vector.

Find the vector field.

One App. One Place for Learning.

Most popular questions from this chapter

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Find vector fields $A$ such that $V = curl A$ for each given V. $V = (y + z) i + (x - z) j + (x^{2} + y^{2}) k$