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Chapter 6: Q21P (page 336)

Find vector fields A such that $V = curlA$ for each given V.
$V = - K$

Short Answer

Expert verified

The vector field is derived to be $A = iy + \nabla u$ .

Step by step solution

01

Given Information.

The given equation is $V = - k$ .

02

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 it magnitude.

03

Find the vector field.

In this problem find the vector A whose curl gives the vector.

$V = - K$

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

role="math" localid="1659242905787" $\frac{\partial A_{z}}{\partial y} - \frac{\partial A_{y}}{\partial z} = 0 \frac{\partial A_{x}}{\partial z} - \frac{\partial A_{z}}{\partial x} = 0 \frac{\partial A_{z}}{\partial y} - \frac{\partial A_{y}}{\partial z} = 0 \frac{\partial A_{y}}{\partial x} - \frac{\partial A_{x}}{\partial y} = - 1$

An obvious solution is mentioned below.

$A = iy + \nabla u$

Here is an arbitrary function.

Hence, the vector field is derived.

$A = iy + \nabla u$

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Most popular questions from this chapter

Find the total work done by forces and if the object undergoes the displacement $C$ . Hint: Can you add the two forces first?

Given

$F_{1} = - 2 yi + (z - 2 x) j + (y + z) k F_{2} = yi + 2 xj$ :

(a) Is $F_{1}$ conservative? Is $F_{2}$ conservative?

(b) Find the work done by 2 on a particle that moves around the ellipse $x = c o s θ$ , $y = 2 sinθ$ from $θ = 0 t o θ = 2 π$

(c) For any conservative force in this problem find a potential function Vsuch

that $F = - \nabla V$ (d) Find the work done by on a particle that moves along the straight line from $(0, 1, 0) to (0, 2, 5)$

(e) Use Green’s theorem and the result of Problem 9.7 to do Part (b) above.

Question:What is wrong with the following “proof” that there are no magnetic fields? By electromagnetic theory,∇· B = 0, and B =∇×A. (The error is not in these equations.) Using them, we find

$\int \int \int \bar{\nabla} \cdot Bdt = 0 = \int \int B \cdot ndσ = \int \int (\nabla \times A) \cdot ndσ = \int A \cdot dr$

Since $\int A \cdot dr = 0$ , A is conservative, or A =∇ψ. Then $B = \nabla \times A = \nabla \times \nabla ψ = 0$ ,so B = 0.

Evaluate each of the integrals in Problems $3$ to $8$ as either a volume integral or a surface integral, whichever is easier.

$∭ \nabla \times Vdτ$ over the volumerole="math" localid="1657334446941" $x^{2} + y^{2} \leq 4, 0 \leq z \leq 5, V = (\sqrt{x^{2} + y^{2}}) (ix + jy)$

$Evaluate the line integral \int (x^{2} - y^{2}) dx - 2 xydy along each of the following paths from (0, 0) to (1, 2) . (a) y = 2 x^{2} . (b) x = t^{2} (c) y = 0 from x = 0 to x = 2; then along the straight line joining (2, 0) to (1, 2) .$

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