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Chapter 14: Q. 3 (page 1153)

Potential Functions: Find a potential function for each vector field.

G(x,y,z)=xzey2i+2xyzey2j+xey2k

Short Answer

Expert verified

Potential function isgx,y,z=x2zey22

Step by step solution

01

Step 1. Given Information.

The Vector field is G(x,y,z)=xzey2i+2xyzey2j+xey2k

02

Step 2. Calculation.

Potential function is given by

gx,y,z=∫xdx+B+C

Here, B is integral with respect to y of the terms in which the x does not appear, and C is integral with respect to z of the terms in which the x and y does not appear.

So,

∫xzey2dx=x2zey22+C.

AndB=0,C=0

Potential function isrole="math" localid="1650887740924" gx,y,z=x2zey22

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Most popular questions from this chapter

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The result of integrating a vector field over a surface is a vector.

(b) True or False: The result of integrating a function over a surface is a scalar.

(c) True or False: For a region R in thexy-plane,dS=dA.

(d) True or False: In computing ∫Sf(x,y,z)dS, the direction of the normal vector is irrelevant.

(e) True or False: If f (x, y, z) is defined on an open region containing a smooth surface S, then ∫Sf(x,y,z)dSmeasures the flow through S in the positive z direction determined by f (x, y, z).

(f) True or False: If F(x, y, z) is defined on an open region containing a smooth surface S , then ∫SF(x,y,z).ndSmeasures the flow through S in the direction of n determined by the field F(x, y, z).

(g) True or False: In computing ∫SF(x,y,z).ndS,the direction of the normal vector is irrelevant.

(h) True or False: In computing ∫SF(x,y,z).ndS,with n pointing in the correct direction, we could use a scalar multiple of n, since the length will cancel in the dSterm.

Find

∫S F(x,y,z)⋅ndSifF(x,y,z)=ln⁡x2+y2+1z+3i+yy+1j+ez2k

Where S is the portion of the sphere with radius 2, centered at the origin, and that lies below the plane with equation z=-2, with n pointing outwards.

Make a chart of all the new notation, definitions, and theorems in this section, including what each new item means in terms you already understand.

F(x,y,z)=−yi+xj−eyzk, where S is the cylinder with equation x2+y2=9from z=2,4, with n pointing outwards.

Use the curl form of Green’s Theorem to write the line integral of F(x, y) about the unit circle as a double integral. Do not evaluate the integral.
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