z=x+iy, consider the entire function f(z)=u(x,y) + iv(x,y) where u(x,y)=3x2y+2x2-y3-2y2. Prove that u is harmonic. Find the harmonic conjugate v of u so that f=u+iv is holomorphic

Q: Let f be a function such that Vf(x, y) = (2x, 1) g(x, y) = x² + y². Find all the points on the curve…

A: Let f be a function such that ∇f(x, y)=2x, 1 and let g(x, y)=x2+y2 We have to find all the points on…

Q: 16) Which function will shift y= vx right 4 units and up 1 unit? A) y = Vx + 4 +1 B) y = Vx + 1 C) y…

A:

Q: The domain of the functi f (x, y, z) = In(3.xy – 6) {(x, y, z) e R³ : xy 20 {(x, y) € R² : xy 2

A: Consider the function  f(x,y,z)=ln(3xy-6)

Q: B- Find the directional derivative of the function W = x² + xy + z³ at the point P: (2,1,1) in the…

A:

Q: analytic function a domain D. If a zeck. y) + b v(x, y) = C in D are real Constants not all o. #2.…

A: If f(z)=u+iv is analytic in a domain D , then f satiesfies the Cauchy Riemann (C-R) equations that…

Q: The conjugate harmonic function v(x, y) of u = In (x² + y²) is

A:

Q: A) Show that the function u(x; y) = 5x4y - 10 x²y³ + y5 is harmonic and determine the harmonic…

A: (A)

Q: how that u(x, y) is harmonic in some domain and find a harmonic njugate v(x, y) hen: u(x, y) = 2x –…

A:

Q: One of the following vector functions is a gradient Select one: O (xy? + 2y?)7+ (2y³ – x²y+ 2x)j O…

A:

Q: Use Lagrange's multiplier(s) to find the maximum and minimum for f(x, y, z) = x² + 2x + 2y2 + 3z2 on…

A:

Q: Prove that the function u(x,y) = x² - y² is harmonic, solute the function v(x, y), so that the…

A:

Q: Show that the function given by f(x,y)= x² + 3xy is differentiable at every poin using the…

A: Introduction: Distinctiveness (differentiability) implies continuity. If a function is…

Q: 1. Obtain the directional derivative of: a.. f(x,y) = x²-4x³y² at the point (1,-2) in the direction…

A: Step 1:

Q: Let f(x, y) = 6x – 2ry and the point M(5, 9). Among all unit directions, there is a unit direction…

A:

Q: Exercise 1: Show that the functions are orthogonal the indicated interval. a) f(x) = x, g(x)=x², x =…

A:

Q: use Euler theorem to show that if f (x, y) is hamog of degree one than x + 0 8 y to fxx (x,y) fyy…

A:

Q: #24 cal Show that f: IR→IR, x-> (1-x)+cos(1+x³) (a) is continous. ay Show that the map 2 f. IR→IR ×…

A:

Q: Determine the voctor ficld of F. F(x, y) = 12i - 4j

A: We have to draw vector field of given vector.

Q: a) Find a scalar function f such that F= Vf where F(x,y) = (3 + 2xy²) i +(2x²y)j.

A: "Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Prove that the directional derivative of f(x, y) = y + 2xy³+x²y2 at point (0, 1) in the direction…

A: The gradient is just the vector of partial derivatives. The partial derivatives of f at the point…

Q: Find the directional derivative of f(x, y, z) = x²y²z + 4xz at (-1,2,1) in the direction 2i + j +…

A: We have to find the directional derivative of f(x, y, z)=x2y2z+4xz at (-1, 2, 1) in the direction…

Q: Q2) For the functions U (x,y)=2x(1-y) and V (x,y) = xy a) Prove that the functions are harmonic. b)…

A:

Q: Show that u (x, y) is harmonic in some domain and |find a harmonic conjugate v(x, y) when y (a) u(x,…

A:

Q: Let f(x,y) be a differentiable function such that f(5,4) = -5. Suppose that points (2,-1,4) and…

A: Tangent plane of the function.

Q: A) Let g(z) = r(x,y) +i s(x, y) is analytic function in the domain D. Show that if Jgl is constant,…

A: