MAT237 Multivariable Calculus with Proofs Problem Set 7
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MAT237 Multivariable Calculus with Proofs
Problem Set 7
Problems
1. Let y : [a, b] → 皿n be a parametrization of a curve C S 皿n , so y is simple and regular. Define L > 0 to be the length of the curve C. Define m : [a, b] → [0, L] to be the arc length parameter of y, so
m(t) = ′at lly/ (u)lldu, a s t s b.
You may assume without proof that lly/ (u)ll is integrable on [a, b]. (Revised 2022-03-12)
(1a) Show that m is continuous on [a, b] and C 1 with m/ > 0 on (a, b). Conclude that m is bijective.
Hint: This is mostly some single variable calculus.
(1b) Show that the inverse map m_1 is continuous on [0, L] and C 1 with (m_1 )/ > 0 on (0, L).
Hint: Start by showing the C 1 property. Then deal with the endpoints. You will need (1a).
(1c) Define g = y o m_1 : [0, L] → 皿n . Prove that g is an arc length parametrization of C .
Hint: You will need (1b) and assumptions on y.
3. Irrotational vector fields are gradient vector fields in some cases.
Theorem A. If U S 皿2 is an open convex set and F is a C 1 irrotational vector field on U, then F is a gradient vector field on U. That is, F = Vf on U for some C2 scalar function f on U.
On the other hand, consider the vector field F(x , y) = ╱ , 、.
(3a) By direct calculation, show that F is irrotational.
(3b) By direct calculation, show that rC (F . T ) ds = 25 where C is the circle (cos t, sin t) for 0 s t s 25.
(3c) Explain why F is not a gradient vector field on its domain and why this does not contradict Theorem A.
(3d) Choose as large as possible of an open set V S 皿2 such that the restriction F lV is a gradient vector field.
You do not need to verify that it is as large as possible, but you should exhibit its potential function.
4. Let F = (f , g) be a vector field in 皿2 with C 1 components f and g. Fix a point p = (x , y) e 皿2 . For 少 > 0, let B少 (p) S 皿2 be the disk of radius 少 centred at p. Orient its boundary a B少 (p) counterclockwise. Do not use Green’s theorem for any part of this question.
(4a) For 少 > 0, show that the circulation of F along a B少 (p) may be expressed as
′ ′ 25
(4b) Since f is C 1 on U, differentiability implies that there exists δf > 0 and Ef : Bδf (0, 0) → 皿 such that V(△x , △y) e Bδf (0, 0), f (x + △x , y + △y) = f (x , y) + a1f (x , y)△x + a2f (x , y)△y + Ef (△x , △y),
where (a,b0,0) = 0. The analogous statement holds for g with δg > 0 and Eg : Bδg (0, 0) → 皿.
Prove that for 0 < 少 < min{δf ,δg }
area(B(1)少 (p)) ′a B少 (p)(F .T ) ds = (curlF )(p)+ 5(1)少 ′025 _Ef (少 cos t , 少 sin t).sin t+Eg (少 cos t , 少 sin t).cos t d t .
(4c) Use the limit definition to prove that 少 5(1)少 ′025 Ef (少 cos t , 少 sin t)sin t d t = 0 and conclude that (curlF )(p) = 少 area(B(1)少 (p)) ′a B少 (p)(F . T ) ds.
2022-03-16