Problem 2. Let N C R² be a bounded smooth domain and p,y E C(N)", Consider a vector field(v1, v2) : N → R² such that v E C'(N) and V · v = 0.V =(1) Prove that(v(x) · Vp(x)) (x) dæ = –-,(v(x) · Vy(x)) p(æ) dæ(2) Assume now that p(t, ·) E C-(N) solves dp+ v · Vp = 0. Show thatdle(t, æ)l²dæ = 0.dt

Answered: Image /qna-images/answer/edd80c1d-dc52-4f53-a712-4fdfc56a3d93.jpg

Answered: Problem 2. Let C R² be a bounded smooth…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter6: The Trigonometric Functions

Section6.6: Additional Trigonometric Graphs

Problem 78E

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9. Sketch the vector field F(x, y) 1 -(yi — хі). Vx? + y?
(7 T Show that e(t) = cos(t) + sin(t), – sin(t) + cos(t)) is the (a) Let F(r, y) = flow line of the vector field. (b) Let F(r, y, 2) = ( , e", 2²) be a vector field. Compute divF and curlF
F(x, y) = x²yi + 3y³x²j Find the integral curves for the vector field а. —6у + 1 = C 6xy2 ob. -6ху — 1 C 6xy2 c. -6ху + 1 = C 6x²y2 O d. -ху + 6 C x?y2
2. Let f : R" → R be a continuously differentiable function, while a E R" is its non-stationary point (i.e., Vf(x) # 0). Moreover, let d* be the vector defined by Vf(x) ||Vf(x)||' d* = - while d is any unit vector in R" different from d* (i.e., d e R", ||d|| = 1, d + d*). Show that there exist real numbers d E (0, 00), ɛ E (0, 0) such that %3D f(x + td*) – f(x + td) 0. (ii) For t z 0, approximate f(x+td*) and f(x+td) using the first-order Taylor polynomial of f(x) at x.
2. Let r = √x² + y² + z². (a) For n ≥ 1 and r > 0, express V(r-n) in terms of r and the radial vector field er. (b) For r> 0, express Vlnr in terms of r and the radial vector field er. (c) Taking A as in question 1, show that for r> 0 we have A(-¹) = 0.
(a) Let F = 2xz²i+j+xy³zk and f(x, y, z) = x²y. Compute the following quantities: (i) Vf (ii) curl(F) (iii) F x Vf (b) Does there exist a vector field G such that curl(G) = (xsin y, cos y, z - xy)? (c) Suppose that F, G : R³ R³ are C¹ vector filelds and f,g: R³ → Rare C² real-valued functions. Prove the following statements: (i) div(F x G) = G curl (F) - F-curl (G) (ii) div(Vfx Vg) = 0 (iv) F. Vf
How do you graph the vector field F = ⟨ƒ(x, y), g(x, y)⟩?
6. Consider the vector field F(ar, y) = (8r°y+ e")î+ (Ka + e")3, where K is some constant. (a) (5 points) For which value of K is F the gradient of a function? For this value, find f such that F = Vf. (b) (5 points) For the value of K found in part (a), evaluate fe F dr, where C is the segment of the curve y = a from (0,0) to (1, 1). (c) (5 points) If instead K = 1, use Green's Theorem to evaluate f F dr, where C is the boundary of the triangle with vertices (0,0), (2,0), and (2, 1), traversed counterclockwise.
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Sketch the vector field F. O F(x, y) = - − ²¹ = 1 + ( y − x) - y
only solute question c ， please

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