5. Determine the divergence and curl of the following vector field and evaluate them at thespecific points:-(a) A=xyzax+4xyay+yzaz, at P(2,-1, 3)(b) B=2pz sin(p) ap +3p z³cos (p) ao, at point Q(5,7/6,2)(c) C-5r cos(0) cos(p) ar +r¹/2; ap, at point S(2, π/4, 7π/4)

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Answered: 5. Determine the divergence and curl of…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Let F(x, y) = ⟨xy^2+2x+1, x^2y − 3y⟩ be a vector field. Let C be the curve thatconsists of two line segments C1 and C2, where C1 has vertices (−1, 0) and (0, 1)and C2 has vertices (0, 1) and (1, 0). C is oriented clockwise.(a) Sketch C accurately.(b) Find the work done by F by moving a particle along the path C in the givendirection.
Find a tangent vector of the curve7(t) = (t², 2 sin(t), 2 cos(t)) at (0,0, 2) (0,0,1) (1,0,0) None of the above or below (0,1,0) O (1,1,1) ) (1/2,0, 1/2)
3. (a) Given the vector valued function r(t) = 4 cos rti + 4 sin rtj + tk. Find the position vector, the unit tangent vector T(t) and the unit normal vector N(t) at t= 1. (b) Given F(r, yz) = 3ry'i + yzj + zrk and o(x, y, 2) = re" sin y. Calculate i. curl curl F ii. V. (Vo) (c) The temperature in degrees celcius at a point (r, y, 2) in a metal solid is ryz T(r,y, 2) =: 1+ 12 + y² + z²° Find the rate of change at the point (1, 1, 1) towards the origin.
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.
F(x,y,z) = ⟨1,y,z^4/3⟩ a. Find the parametric equation of a curve passing through the point (-1,1,1) which is parallel to the vector field F at all of its points, with the parameter t = x. b. Find the equation of a surface passing through the point (-1,1,1) which is perpendicular to the vector field F at all of its points.
Suppose F(x, y) =-yi+aj and C is the line segment from point P = (3,0) to Q = (0,4). (a) Find a vector parametric equation r(t) for the line segment C so that points P and Q correspond to t= 0 and t = 1, respectively. F(t) = (b) Using the parametrization in part (a), the line integral of F along Cis So - [* F(F(t)) - F' (t) dt = SC dt with limits of integration a = and b= F. . dr = (c) Evaluate the line integral in part (b). (d) What is the line integral of F around the clockwise-oriented triangle with corners at the origin, P, and Q? Hint: Sketch the vector field and the triangle.
Use the rule of differentiation of the scalar product, to evaluate(u · v) for the following vector fields: and a) ú= Calculate the time derivatives of u and v. Represent your answers in the form i+ j+ k and v = i+ (u. v)= j+ u = eªt i + 4t² j +1 k. е k. v = 5i + sin(3t)j + cos(6t) k. b) Now calculate the time derivative of scalar product of u and v: d (u dt

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5. Determine the divergence and curl of the following vector field and evaluate them at the specific points:- (a) A=xyzax+4xyay+yzaz, at P(2,-1, 3) (b) B=2pz sin(p) ap +3p z'cos (9) ao, at point Q(5,7/6,2) (c) C=5r cos(0) cos(p) ar +r¹/2 a, at point S(2, π/4, 7/4)