Q2) A) Verify that wxy = Wyx for w = x sin y + y sin x+ xy. B) Show that f(x, y) = e-y cos 2x satisfies a Laplace Equation C) Find the derivative of f(x, y, z) = In(2x + 3y+ 6z) at (-1,-1, 1) in the ょ,つょ !!

Q: 4. Find the work done by the force field F (x,y) = (x + y)î + (xy)ĵ in moving a particle along the…

A: The force field is given by F→ = x+yi^+xyj^ The end points of the straight line along which the…

Q: Find the gradient vector field for the scalar function f(x,y)=sin(2x)cos(6y).   Enter the exact…

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Q: b) Show that F - dr is independent of its path, given that y F(r, y, 2) = 1+. + tan-2) i+ tanrj+ k.…

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Q: (b)Find the derivative of g in the direction of a= 2i+ curl w where at point (0,-1,1) g = In- 4 + 2…

A: Given, w=cosxyzi+xeyz+1j+xyzk Then, Curl…

Q: 8. Find the flux of F = (2x – 5y², 3y, x) out of the sphcre of radius 2 centered at the origin.

A: The flux of the vector space is given by the formula, ∫S∇.F dS where S is the sphere of radius 2…

Q: þ/ Show that the function satisfy Laplace equation f(x,y) = In /9xy+3y² Q/ find by using chain rule…

A: Since you have posted multiple questions ,we solved the one question for you  To get the solution of…

Q: Show that if the function f(x, y) = ex siny + ecosx, satisfies Laplace Equation.

A: We can solve this as follows:

Q: In(xy) cos(Tx + T) when x 2 and y = 1/2.- Find the tangent plane to f(x, y)

A: We have to find the equation of  tangent plane.

Q: Find the gradient vector field (F(x, y, z)) of f(x, y, z) = ln(x + 5y + 6z). F(x, y, z) = =

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Q: 10. Does the function f (x, y, z) = (x² + y2 + z?)-1/2 satisfy Laplace equation?

A: The objective is to check the given function satisfies Laplace equation or not.

Q: Q2) A) Verify that wxy wyx for w x sin y+y sin x +xy. %3D R) Show that f(x v) = e-2y cos 2r…

A: See the details solution in below

Q: Find the conservative vector field for the potential function by finding its gradient. f(x, y) =…

A: It is required to find the conservative vector field for the potential function given in the…

Q: w In(x2 + y2 +z?), x = ue" sin u, y = ue" cos u and z ue", using chain rule at the point (u, v) =…

A: Given w=lnx2+y2+z2x=uevsinuy=uevcosuz=uev

Q: Show that the function z = arctan y/x satisfies Laplace’s equation                           ∂2z/…

A: Consider the following equation:

Q: a) Evaluate the line integralZ F dr C in terms of t where F = cosyi+xj+ye²>k and C is the line…

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Q: (b) Find the derivative of g in the direction of a=2i+curl w at point (0,-1,1) where u u +2 g= In. J…

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Q: 2. Calculate the directional derivative of the scalar field U(x, y,z) at the point M,, along the…

A: Given that : U=xz3y-x3y=xz3y-x32y12 The gradient will be calculated as :…

Q: 1. Give an equation of the tangent plane to the surface z = In(4y – x) at the point (3,1,0). 2. Use…

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

Q: Evaluate line integral of vector field F(z,9) = i+ 1- x? y2 2x along the curve x = cos(t), y =…

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Q: 4. Find the flow of the velocity field F = V(xy²z³) along the line segment from (1,1,1) to (2,1,…

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Q: FIND THE DIVERGENCE OF THE VECTOR F GIVEN = sin (8x)i + cos (5y); + z ² k FIELD F(X, Y, 2) BY

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Q: Show that the displacement u(x, t) in an elastic string of length t fireed at both ends which is set…

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Q: Find the gradient field associated with the function φ(x, y, z) = xyz.

A: To find the gradient field associated with the function  φ(x, y, z) = xyz.

Q: Find the work done in moving a particle in the force field F : 3x²i + (2xz - y)j + zk along the…

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Q: Evaluate the line integral f (-y sin(xy)i – x sin(xy)j) dr using the potential function f(x, y) =…

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Q: 10. Show that the function f(x, y, z) = e3z+4y cos 5z satisfies the Laplace equaton. Good Luck.

A:

Q: 2. Integrate f(x, y) = sin /x2 + y² А. over the closed unit disk.

A:

Q: Compute the directional derivative of f(r. y) = = at the point (1, 1) in a irection normmal to the…

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Q: The direchional derivative of the salan funchion at (l,-1,0) in the direction (i-j+2k) is xy- z- xyz…

A: Given that  Let f(x,y,z)=x^2y-y^2z-xyz To find the directional derivative at point

Q: Find g(x, z) so that F(x, y,z) = (8x cos y sin? z + 2e2xz)î – (4x² sin y sin? z)j+ (g(x, z) cos y +…

A: Given that    Fx,y,z=8x cosy sin2z+2e2xz i^-4x2 siny sin2z j^+gx,zcosy+2e2xz k^ Now we have to find…

Q: Find the conservative vector field for the potential function by finding its gradient. f(x, y, z) =…

A: Given: f(x, y, z) = 9x2 − xy − z2

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: Find the divergence of F(x, y, z) =(a -1 Y, y? + 1' 3xz2 + 2y ) at (2,0, – x COS

A:

Q: Find the derivative of g in the direction of a = 2i+ curl w at point (0 where น u+2 8 = In- J u =…

A: Given: the function g=lnuu+2, u=z(xy+z2)2, w=cosxyzi+xeyz+1j+xyzk To find: the directional…

Q: Show that if the function f(x, y) = ecos-e-cosx, satisfies Laplace Equation.

A:

Q: The linear approximation to f(x, y) = /2y + cos2(x) at the point (0,0) takes the form Ax + By + C.…

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Q: Show that the function f(x, y) = In /x² + y² satisfies the Laplace equation. %3D

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Q: Let F(x, y, z) = (sin(xy), cos(xy), sin(æyz)). What is the divergence of F at the point (2, –2, 27)?…

A: Given F→=sinxy, cosxy, sinxyz and a point 2,-2,2π

Q: Find the conservative vector field for the potential function by finding its gradient. f(x, y,…

A:

Q: Consider F(x, y, z)=e-2 yi-y³/²zx²j+ln(22)k and calculate the divergence of F.

A: Disclaimer: Since you have asked multiple questions, we will solve the first question for you. If…

Q: Verify whether the function f(z) = e^x (cos y + isin y) satisfies Cauchy- Riemann equations or not.

A: We know that cauchy-Riemann equations  ux = vy  and uy = -vx Here  u = excosy and v = exsiny

Q: Q2) A) Verify that wxy = Wyx for w = x sin y + y sin x + xy. B) Show that f(x, y) = e-2y cos 2x…

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Q: b/ Show that the function satisfy Laplace equation f(x,y) = In /9xy+ 3y² %3D

A: Since you have asked multiple questions in a single request so we will be answering only first…

Q: .a. Compute the divergence of the function (V. v) v = (r² cos²0) î + (r² sin²0)@ + (r sin® cosø)@

A: Note: Since you have asked multiple questions, we will solve the first question. Please resubmit the…

Q: 10. Find the work done by the force F=vx +2y y - 7 for a particle that moves between points (-1,0,1)…

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Q: 2: Compute the work done by the force field F = (y, -x) to move an object from point A = (1,0) to B…

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Q: Prove that the directional derivative of flay.2) = aysinz. direction v=i +2j+2k 1Is Sin Z at…

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Q: how that the function satisfy Laplace equation f(x,y) = In /9xy+3y²

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Q: Let F(r, y, z) = (y² + 2z, 2ry + 4, 2a), and let the curve C be described by x = t, y = t, z = t3…

A: