Show that u(x, y) = e³ cos x is harmonic and find the harmonic conjugate v such that v (0,0) = 0.

Q: Find all harmonic conjugates of u(x, y) = x³ - 3xy² + 2y.

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Q: Calculate for w-z sin(xy+).

A: The given function is w=zsinxy4+9z

Q: (Yp)for y2) + 4y = 3cos2x ;is %3D اختر احدى الدجابات a. y = c, cos2x + C2sin2x b. y x2(Acosx +…

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Q: Evaluate r(2) and r(-1) for r(t) = (sin 1,12, (1² + 1)~1). %3D

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Q: The value of 'b' for which u(x, y) = ebx cos 5y is harmonic

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Q: c) Let f(x, y,z) = cos (xyz)sin(x²y=

A: The given function is f(x,y,z)=cos2(xyz)sin(x2y2z2) ..............(1) we have to find;  fx=∂f∂x=?and…

Q: Show that the function u = In(X² + Y2) is harmonic and find the conjugate harmonic v.

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Q: Encuentre TLt), NCt) B (t) de cLE) = ( cost , sent, t)

A: We will use the respective definitions.

Q: Find the points of intersection between r =2 cos20 andr=- %3D 1- cose

A: To find the intersection point two curve we have to find r,θ by solving given curve.

Q: Find V (V x F) if F(x, y, z) = 8 sin xi + 6 cos(2.x - 3y)j + 7zk. %3D | V. (V x F) =

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Q: Prove that sinh z = sinh x cos y +i cosh r sin y.

A: To prove, sinh z = sinh x * cos y + i cosh x * sin y