For number 9The answer is ACould you show me how equals:(A) 0(B) 2πi/9(C) -2π/3-2πi/9(D) 1/99 The outward flux of the vector field F(x, y, z) = (1/2 + e+e+z(A) 2πthe boundary of the region E = {(x, y, z) = R³: x² + y²+z2 <4, y ≥0, z> 0} is22, ex+z+e+y2through(B) T(C) -π(D) 2√210 Consider the vector field F(x, y, z) = (±±±², 5, +2²)ThenN =defined on the set= {(x, y, z) = R³:-5

Answered: Image /qna-images/answer/d20764b5-9211-4949-a1fd-b009bfec5253.jpg

Answered: equals: (A) 0 (B) 2πi/9 (C) -2π/3-2πi/9…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Let (4xy-3x²z²)₁ + (2x²)Ĵ − 2x³ zk = (i) Find a scalar potential for F. (x, y, z) = - note: Answer should have "+C" in it, where C is some constant (ii) Find the work done in moving from (1,1,0) to (2, −1, 1) by the above force field. W= (iii) Solve the differential equation (4xy - 3x²²)dx + 2x²dy - 2x³zdz = 0 ☐ = K, where K is a constant
A fluid flow in R³ has velocity field i = -x?î + xj + 2zk. (a) Find the equation of the field line which passes through an arbitrary point (xo, Yo, 2o) at time t = 0. (b) A fluid particle is at position (1,0, e²) at time t = 0. Find its position at time t = 1.
The gradient vector field of f(x,y)=cos(x) - 2sin(y) is: O1. f(x,y)= cos(x) i - 2sin(y) j II. f(x,y)=-sin(x) i - 2cos(y) j III. f(x,y)= sin(x) i - 2cos(y) j IV. f(x,y)=-sin(x) i+ 2cos(y) j
Cal 3
Determine whether or not the given vector field F is conservative. If so, find a potential function. (a) F(x, y) = (x, y) (b) F(x, y) = (3x²y, x³ (c) F(x, y) = (2xe" + y, x² e +x - 2y) (d) F(x, y) = (siny - y sin x+x, cos x + x cos y + y) (e) F(x, y) = (sin(xy) + xy cos(xy), x² cos(xy)) (f) F(x, y, z) = (x, y, z) - (g) F(x, y, z) = (x + 2, −y - z, x − y) (h) F(x, y, z) = (2xy³, x²z³, 3x²yz²) (i) F(x, y, z) = (3y4z2, 4x³ z², -3x²y²) (j) F(x, y, z) = (2x² + 8xy², 3x³y - 3xy, -4y² z² - 2x³z) (k) F(x, y, z) = (y² cosx+23, -4+ 2y sin x, 3xz² + 2) (1) F(x, y, z) = (4xy - 3x²x² + 1,2x² + 2, −2x³z+3z²) Linear vector fields (a) On R² with coordinates x, y, define a vector field to be linear if it is given by a matrix formula = M + V a = -635+6-2+3+; C M = ax+by+e [cx + dy+f] Find necessary and sufficient conditions for to be conservative. Prove that the conditions you wrote are correct. (b) Repeat for a vector field in R3 (using a 3 × 3 matrix in place of M and a 3 × 1 column vector in place of…
(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
a) Find the values of a,b,c which make f(x,y,z) = sin (xªy°z©) a potential function for the vector field F(x,y,z)= 2 x'y®z4 cos (x?y®z4) i+ 5x²y^z* cos (x?y5z4) j + 4 x?y5z³ cos (x²y®z4) k a= b= C= b) Find the work of F along the curve r(t)=(2t, e (t - 1), e t(t – 1) for 0 sts1. Do not approximate your answer. For example, leave sin (e ) as that, not 0.41. Work = F•dr =
3. Consider the vector field F(x, y) = (3y²+ 4x, 5y sin(x)). (a) Calculate curl F at the point (0, 1). (b) Explain the meaning of the number you found above. What does the sign of this number tell us? (c) Is the vector field F(x, y) = (3y²+4x, 5y sin(x)) conservative? Justify your answer.
4. Match the vector fields given below with their plots. (a) F(x, y, z) = (b) F(x, y, 2) = (c) F(x, Y, z) = x² + y² (d) F(x, y, z) = (x² + y² + z²)3/2’ (x² + y² + z²)3/2' (x² + y² + 2²)3/2 Field (I) Field (II) Field (III) Field (VI)

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