1. (18 pts.) Short answer: a) Given that f (u,v,w,x, y,z) is a real valued function,...
Question 6 6 pts Suppose that f(x, y, z) is a scalar-valued function and F(x, y, z) = (P(x, y, z), Q(2,y,z), R(x, y, z)) is a vector field. If P, Q, R, and f all have continuous partial derivatives, then which of the following equations is invalid? O curl (aF) N21 = a curl F for any positive integer Q. REC o div (fF) = fdiv F+FVF Odiv curl F = 0 O grad div f = div grad...
Consider a real-valued function u(x, y), where x and y are real variables. For each way of defining u(x, y) below, determine whether there exists a real-valued function v(x, y) such that f(z) = u(x, y) + iv(x, y) is a function analytic in some domain D C C. If such a v(x, y) exists, find one such and determine the domain of analyticity D for f(z). If such a v(x, y) does not exist, prove that it does not...
Q7 Prove the real valued function in x and y given by 1) and (ii) are harmonic. Find the corresponding harmonic conjugate function and hence construct the analytic function f(z) = u(x,y) +j v (x,y) 0v(x, y) = In(y2 + x2) + x + y, z = 0 (ii) u(x,y) = y2 – x2 + 16xy
Please try helping with all three questions.......please 1 point) Integratef(x, y, z) 6xz over the region in the first octant (x,y, z 0) above the parabolic cylinder z = y2 and below the paraboloid Answer Find the volume of the solid in R3 bounded by y-x2 , x-уг, z-x + y + 24, and Z-0. Consider the triple integral fsPw xyz2 dV, where W is the region bounded by Write the triple integral as an iterated integral in the order...
35. f(z) (3z+5)5 f(x) = SK-53 x-2 x2 -2x 37. f*)-+1 In problems 54-61, find dy/dx. 54. xy'- xy+10 0 In problems 62-63, find the equation of the tangent line to the given graph at the given point. 62. yxy - 6 0 at the point (1,2) 63. x+xy - vy-3 0 at the point (1,4) In problems 64-78, find y for the equation. 35. f(z) (3z+5)5 f(x) = SK-53 x-2 x2 -2x 37. f*)-+1 In problems 54-61, find dy/dx....
5] (2) GIVEN: a> 0,0# {(x, y, z) z a"-x'-y") W is the solid region of R' that is below 2 and above the xy- plane. W has constant density,8 and the mass of W is M, m(W) M FIND: The moment of inertia, I, of W with respect to the z- axis, express 2 I in terms of M and a without 8
Consider the real valued function of two variablesz-f(x,y), which satisfies Ar2 +4y2+ 224 (a) Describe all points (ro. yo. f(ro. yo)), such that the tangent plane to f(x, y) at (o.yof(ro. yo)) the point (2,0,0). Include scription. (b) Describe all points (roo,f(o, yo)), such that the tangent plane to f(x, y) at (zo. yo. f(o. yo) passes through the point (2,0,27), where >0 is an arbitrary constant. With the aid of a computer or by hand, provide a sketch of...
Consider the following function 6 f(x, y,z)=z - x? cos(my) + xy? (i) Find the gradient of the function f(x, y, z) at the point P,(2,-1,-7). (ii) Find the directional derivative of f(x, y, z) at P,(2,-1,-7) along the direction of the vector ū = 2î+j+2k. (iii) Find the equation of the tangent plane to the surface given below at the point P,(2,-1, -7). 6 :- xcos(ty) + = 0 xy
2- a) The real part of a complex function f(z) given as, u(x, y) = 3x?y - y. Iff(2) is an analytic function, find v(x,y) and f(z) (15p) b) Find the whether f(z) is analytic or not where f(z) = cos(x) +ie'sinx. (15p)
410. [V] The transformation T.1.1: R3 R3, Tk,1,1 (u, v, w) = (x, y, z) of the form x = ku, y = 0, z = w, wherek #1 is a positive real number, is called a stretch if k > 1 and a compression if 0 <k < 1 in the x-direction. Use a CAS to evaluate the integral e-(4x2+9y?+252) dx dy dz on the solid S = {(x, y, z)|4x² +9y2 + 25z< 1} by considering the compression...