MATH 132 Exercise Ch12 PDF

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CHAPTER 42

CLOSED AND EXACT DIFFERENTIAL FORMS 42.1. Background Topics: closed differential forms, exact differential forms. 42.1.1. Definition. A k-form ω is closed if dω = 0. It is exact if there exists a (k − 1)-form η such that ω = dη .

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42. CLOSED AND EXACT DIFFERENTIAL FORMS

42.2. Exercises (1) The 1-form ω = ye where f (x, y, z) =

xy

dx + xe

xy

dy

(is/is not) exact. If it is then ω = df .

(2) The 1-form ω = x sin y dx + y cos x dy (is/is not) exact. If it is then ω = df where f (x, y, z) = .     arcsin x x arctan y x2 2 y dx + dy (is/is + + 3x (3) The 1-form ω = √ − 2 +e y 2y 1 + y2 1 − x2 not) exact. If it is then ω = df where f (x, y, z) =

.

2x3 y + 2xy + 1 1 + x2 exact. If it is then ω = df where

(4) The 1-form ω =





dx + (x2 + ez ) dy + (yez + 2z) dz

f (x, y, z) =

(is/is not)

.

(5) The 1-form ω = (yzexyz +2xy3 ) dx +(xzexyz +3x2 y2 +sin z) dy +(xyexyz +y cos z +4z 3 ) dz (is/is not) exact. If it is, then ω = df where f (x, y, z) =

.

(6) Solve the initial value problem ex cos y + 2x − ex (sin y)y′ = 0,

y(0) = π/3.

Hint. Why is the 1-form (e cos y + 2x) dx − e (sin y) dy exact? Answer: y(x) = . x

x

(7) Solve the differential equation 2x3 y2 + x4 y y′ = 0 on the interval 1 ≤ x ≤ 10 subject to the condition y(2) = 12 . Hint. Is 2x3 y2 dx + x4 y dy exact? Answer: y(x) =

.

42.3. PROBLEMS

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42.3. Problems (1) Show that every exact k-form is closed. (2) Show that if ω and µ are closed differential forms, then so is ω ∧ µ.

(3) Show that if ω is exact and µ is closed, then ω ∧ µ is exact.

(4) Show that if the 1-form ω = P dx + Q dy + R dz is exact, then Py = Qx , Pz = Rx , and Qz = Ry . (5) Suppose that F is a smooth vector field in R3 and that ω is its associated 1-form. Show that ∗ dω is the 1-form associated with curl F. (6) Let F be a vector field on R3 and ω be its associated 1-form. Show that ∗ d ∗ ω = div F.

(7) Let f be a smooth scalar field (that is, a 0-form) in R3 . Use differential forms (but not partial derivatives) to show that curl grad f = 0. (8) Let F be a vector field on an open subset of R3 . Use differential forms (but not partial derivatives) to show that div curl F = 0. (9) Use differential forms to show that the cross product of two irrotational vector fields is incompressible (solenoidal). Hint. Show (without using partial derivatives) that div(ω × µ) = curl ω · µ − ω · curl µ.

(10) Explain how you know that there does not exist a vector field defined on R3 whose curl is yz 2 i + x4 yz j + y2 z k.

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42. CLOSED AND EXACT DIFFERENTIAL FORMS

42.4. Answers to Odd-Numbered Exercises (1) is, exp(xy) (3) is, arcsin x arctan y +

x2 + x3 + ey 2y

(5) is, exp(xyz) + x2 y3 + y sin z + z 4 2 (7) 2 x...