It is easy to see that u(z) = Im (+) is harmonic in the unit disk |2| <1 and lim,→1- u(reio) = 0 for all 0. Why does this not contradict the maximum principle for harmonic functions? Is u continuous on |2| = 1?

Q: Verify divergence theorem for f=x2i+zi+yzk taken over the cube bounded by x=0,x=1,y=0,y=1,z=0 and…

A: Use the divergence theorem to solve this problem.

Q: V.(V×F)= 0

A: Now given, ∇.∇×F=0 To prove the above.

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Q: Find the tangent plane for the level surface f(x,y,z) = 5x4 + cos (4y) + In (6z) = c at the point…

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Q: Evaluate both side of divergence theorem for cube define by -0.1< x. y.z<0.1 if D = 6xe2Y (āx + xay)

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Q: what is the area of the rotational surface formed by rotating the part of the function y= x^3/3…

A: We apply the above formula to solve this

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A: A function ux,y is called harmonic if it is twice continuously differentiable and satisfies the…

Q: | Use Gauss Divergence theorem to calculate (5zi+ 4yj - 2: k).n dA over the sphere +y+ 9.

A: Here, let F=5x,4y,-2z⇒div F=∂(5x)∂x+∂(4y)∂y+∂(-2z)∂z=5+4-2=7

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Q: If f: (a,b]-R has a local minimum at pe (a,b) and f is differentiable at p, then prove that f'(p) =…

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Q: Show that f(z) = x ² + y ² where z = x + iy is not analytic anywhere using Cauchy Riemann equations.

A: Cauchy Riemann Equations: Given,  f(x+iy)=u(x,y)+iv(x,y)  and it should satisfy  ∂u∂x=∂v∂y and…

Q: Prove that af +y. -yof=nf (x, y). əx (Hint: Differentiate both sides of the definition of a…

A: Given  a) x∂f∂x+y∂f∂y=nf(x,y)b) x2∂2f∂x2+2xy∂2f∂x∂y+y2∂2f∂y2=n(n-1)f(x,y)

Q: Suppose that Di is the open disk with center 0 and radius 1. Let f(z) be analytic on Di \ {0} and…

A: Given- Suppose that D1 is the open disk with center 0 and radius 1. Let fz be analytic on D1\0 and…

Q: Find the local linearization of f(z) =z² at 9. lo(z) = %3D

A: Given : f(x) = x2   x0 =  9

Q: 3. If C is the unit circle in the complex plane C, and f(2)= =², show that f(z) dz = 0 using two…

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Q: Q3: Prove that the divergence of a curl is always zero.

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Q: 6. Let f(x,y) = sin(x + y) + arcsin (xy) a. Find the linearization L(x,y) at the point (0, ")

A: We have to find linearization of given function f(x,y): Concept: The linearization of a function…

Q: (15) Find the divergence and curl of i = (ryz)î + (3x²y)ĵ + (æz² – y²z)& at point (2, –1,1).

A: Given v→=xyzi^+3x2yj^+xz2−y2k^ We have to find the divergence and curl of v→ at point 2,−1,1. If…

Q: Use the Cauchy-Riemann Equations to show that f(z) = z Im(z) is only differentiable at z =0 and find…

A: Cauchy-Riemann Equations: Let f=u+iv be a complex valued function , then the Cauchy-Riemann…

Q: 4. Let f(z) = 1 23 1 - and find [ f(z) dz, where C a circle of radius 2 about the origin with…

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Q: 4.) Verify the Divergence Theorem where F(x, y, z)=(x',xz,3z) given by x' + y +z² <4. and E is the…

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Q: Find the tangent plane for the level surface f(x,y,z) = 3x5 + cos (5y) + In (5z) = c at the point…

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Q: Suppose that (z) is entire and that the harmonic function u(x, y) = Re[fz)] has an upper bound uo;…

A: As requested following is the answer for question 12.

Q: 4. (Section 17.8) Use the Divergence Theorem to calculate [[F F-dS, where F = y + ryj – zk and S is…

A: To use the divergence theorem to calculate ∫∫SF⋅dS.

Q: Evaluate both sides of divergence theorem for A = pa, + pcos()a,+ za, through the closed surface…

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Q: Verify the divergence theorem for F= 3i+xyj+xk taken over the region bounded by z = 4-y^2 , x=0 and…

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Q: Prove that the function f(z2) defined by x*(1+ i) – y°(1 – 1) , (z ± 0) x² + y? f(z) = [0, (z = 0)…

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Q: 5. Compute / | F. aš, where F = (y, -x, 2z) and S is the surface of the sphere of radius 2, centered…

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Q: 4. Let f(x, y) be a function defined on a disk D not containing the origin which is given by 2.x2 f…

A: as per our company guideline we are supposed to answer only one qs kindly post remaining qs in next…

Q: (b) Show that the function defined by IN if (x, y) (0,0), √√√z²+y² f(x, y) = 0 if (x, y) = (0,0) is…

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Q: A necessary condition for the functional defined by X2 = dx f(x, Y(x),Y'(x)) to have a local minimum…

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Q: Apply the theorem of Cauchy-Riemann equations to verify that function f (z) is entire: f (z) = 3x +…

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Q: Evaluate both sides of divergence theorem for A= pa,+ pcos()a+ za, through the closed surface…

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Q: Find the tangent plane for the level surface f(x,y,z) = 7x5 + cos (5y) + In (9z) = c at the point…

A: Answer: In oerder to use gradients we introduce a new variable c= 7x5+cos(5y)+ln(9z) Therefore the…

Q: If F = (x² − y² + 2xz)i + (xz − xy + yz)j + (z² + x²)k, find the divergence of - curl F and curl of…

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Q: Supose z is implicitly defined by the eq qy+ 3x*+ XyZ + 3Z=9 in an emvoment of the point P(0,3) in…

A: In this question we will use basic of vector calculus i.e concept of directional derivative. We will…

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Q: Evaluate both sides of divergence theorem for A = pa, + pcos($)a+za, through the closed surface…

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Q: Apply Gauss Divergence theorem for F = x²i + y²j + z²k_over the cuboid S bounded by 0 < x< 1, 0< y<…

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Q: Evaluate both sides of divergence theorem for A = pa,+ pcos(ø)a,+ zaz through the closed surface…

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Q: ) Show that the function defined by if (x,y) = (0,0), f(x, y) = { 0 if (x, y) = (0,0) is not…

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Q: Consider a potential problem in the half-space defined by z 0, with Dirichlet boundary conditions on…

A: The green function solution to the potential problem with this boundary for Dirichlet boundary…

Q: Evaluate / (7xi – zk) · ndA over the sphere S : x² + y² + z² = | S a) by divergence theorem, (b)…

A: a) By divergence theorem, ∬SA. ndA=∫∫∫V∇.A dV Now,  ∇.A=∂7x∂x+∂0∂y+∂-z∂z=7-1=6 By changing to…

Q: Evaluate both sides of divergence theorem for A= pa, + pcos(0)a , + zaz through the closed surface…

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Q: 8. Let F = 2 (in 2D). Show that F is divergence-free on its domain, that is, V.F= 0 at all points…

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Q: Show that f (z) = e? is not analytic using Cauchy-Riemann equations.

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