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Chapter 6: Q18P (page 295)

As in Problem 17, find the following gradients in two ways and show that your answers are equivalent ∇x.

Short Answer

Expert verified

The solution to this problem is∇x=i^.

Step by step solution

01

Given Information.

The given information is ∇x.

02

Definition of Scalar field.

In mathematics and physics, scalar field or scalar-valued function referred to a scalarvalue to everypointin aspace– possiblyphysical space. The scalar may either be a (dimensionless)mathematical numberor aphysical quantity.

03

Find the solution.

Usetheequation∇f=er∂f∂r+eθ1r∂f∂θ+ez∂f∂z∇x=∇rcosθ=∂rcosθ∂rer+∂rcosθ∂r1reθ∇x=cosθer-sinθeθSimplifyfurther.∇x=cosθcosθi^+sinθj^-sinθ-sinθi^+cosθj^=i^Hence,thesolutiontothisproblemis∇x=i^.

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Most popular questions from this chapter

Hint:Integrate(g)Derive the following vector integral theorems

(a) ∫volumeτ∇ϕdτ=∮surfaceinclosingτϕndσ

Hint: In the divergence theorem (10.17), substitute V=Ï•°äwhere is an arbitrary constant vector, to obtain C⋅∫∇ϕdÏ„=C⋅∮ϕndσSince C is arbitrary, let C=i to show that the x components of the two integrals are equal; similarly, let C=j and C=k to show that the y components are equal and the z components are equal.

(b) ∫volumeτ∇×Vdτ=∮surfaceinclosingτn×Vdσ

Hint: Replace in the divergence theorem by where is an arbitrary constant vector. Follow the last part of the hint in (a).

(c) localid="1659323284980" ∫curveboundingσϕdr=∮surfaceσ(n×∇ϕ)dσ.

(d) ∫curve²ú´Ç³Ü²Ô»å¾±²Ô²µÏƒÏ•dr×V=∮surface(n×∇)×Vdσ

Hints for (c) and (d): Use the substitutions suggested in (a) and (b) but in Stokes' theorem (11.9) instead of the divergence theorem.

(e) ∫volumeτ∇ϕdτ=∮surfaceinclosingτϕV·ndσ-∮surfaceinclosingτϕV·∇ϕndτ.

Hint: Integrate (7.6) over volume and use the divergence theorem.

(f) localid="1659324199695" ∫volumeτV·(∇×∪)dτ=∫volumeτV·(∇×∪)dτ+∮surfaceinclosingτ(∪×V)·ndσ

Hint: Integrate (h) in the Table of Vector Identities (page 339) and use the divergence theorem.

(g) ∫surfaceofσϕ(∇×V)⋅ndσ=∫surfaceofσ(∇×∇ϕ)⋅ndσ+∮curveboundingϕV⋅dr

Hint:Integrate(g)in the Table of Vector Identities (page 339) and use Stokes' Theorem.

Evaluate each of the following integrals in the easiest way you can.

∮(2ydx-3xdy)around the square bounded by x=3,x=5,y=1andy=3.

Given Ï•=x2-yz and the point (3,4,1) find

(a) ∇φ at P ;

(b) a unit vector normal to the surface φ=5 at P ;

(c) a vector in the direction of most rapid increase of φ at P;

(d) the magnitude of the vector in (c);

(e) the derivative of at in a direction parallel to the liner=i-j+2k+(6i-j-4k)t.

∬VײԻåσover the entire surface of the cube in the first octant with three faces in the three coordinate planes and the other three faces intersecting at (2,2,2), where V=(2−y)i+xzj+xyzk.

Given Ï•=z2-3xy find

(a) grad role="math" localid="1659325059343" Ï•;

(b) The directional derivative of Ï•at the point role="math" localid="1659325089841" (1,2,3) in the directionrole="math" localid="1659325033087" i+j+k;

(c) The equations of the tangent plane and of the normal line to Ï•=3 at the point (1,2,3)
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