Show that the free space Green function in two dimensions has the form G(p,p') = -21n|p - p' by integrating 1/R with respect to (z - z') between the limits +7 where |Z| » 1. For a cylinder with radius a and the specified potential (a, p') on the boundary, find the Green function and the expression for the potential inside the cylinder.

Q: 2. A ball is rolling around on a surface with shape given by z = f(x, y) = e-5y sin²x. The ball…

A: (a) To calculate the direction in which the ball would roll from a general point (x, y), we need to…

Q: 11. Integrate the function. y? -dy (49 – y² ) 5 y 7y +C - sin"' y + C V49 – y V49 – y² V49 - y -…

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Q: x3 dx (x – 1)5 y = Select one:

A: First evaluate the integral by substitute method. Then apply the limits of integral.

Q: Find a potential function for F. f(x,y,z) =

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Q: Find y' for x At (1, 2), y' 3 + Y 5 Sutinent 33 32 at the point (1, 2).

A: To find y' for the given function

Q: 4xy* :- find f,,f, at point (4,1) for f(x,y) = x’e**Y

A: I have used partial differentiation

Q: Draw the planar representation of the graph below

A: Given is the graph. The objective is to represent it in planar form, that is,to draw it in such a…

Q: 4. Compute the first fundamental form.

A: Given xu,v=u2-cosv,u+sint, cosv

Q: aw aw = f(s³ + t²) and f'(x) = e*. Find and at əs Assume that w=f[s]

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Q: A cylindrical vessel without lid is 4 cm in radius and 10 cm in height. If the vessel is to be…

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Q: k xy' - y = x*y"

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Q: Sketch the slope field and solution curves   a) yy' + 4x = 0, (1,1), (0,2)

A: The given equation is yy'+4x=0 for 1, 1 and 0, 2. To find: sketch the slope field and solution…

Q: Find a potential function for F(x, y, z) = :yi+xj+xzk or prove none exists.

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Q: Calculate the frist order differential of the function f(x,y) = x^{4}*y^{2}+2*x-y^{7}+3

A: First order Partial derivatives

Q: Find the work done by F(x, y) = (x, Y) over the curve r(t) = (e' cost, e' sin t) (x² + y?)3/2 from…

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Q: 1 1 1 1 ++* 1 2 1 1 -2 1 1 Which system of ODEs matches the velocity field above? Ox'= 1, y = x…

A: The velocity field is shown in the picture. Consider, x'=yy'=1 This means: y'x'=1yydy=dx Integrate…

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: Q4 A) Consider the i.v.p x' = t^(2) + cos(x), x(0) = 0. Verify that the hypothesis of Cauchy Picard…

A: The objective of the question is to verify the hypothesis of the Cauchy Picard theorem for the given…

Q: Consider a fully developed two-dimensional channel flow between two infinite parallel plates…

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Q: Let z be a differentiable function of x and y related by the following equation: x+z (√y + cosh(z))…

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Q: Find the potential function for F(x,y,z) = (x^-1z)i + (y^-1z)j + ln(xy)k. Show all steps and explain

A: Potential function of  F, is f such that ∇f=F.

Q: find fsub(x), fsub(y), and fsub(z): f(x,y,z)=e^yz+sinx)

A: Given,          fx,y,z=eyz+sinx

Q: The magnetic field B due to a small current loop (which is placed at the origin) is called a…

A: Let S be a surface enclosed by the curve C , then ∬ScurlF→⋅dS→=∫CF→⋅dc→

Q: the functional F[y] = [[(y^)² - y² + 4ye™ ]dx such that y(0) = y(7/2) = 0. Calculate the minimal…

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Q: The area bounded by y=1-x² and x+y=1 is submerged vertically in the water with the water level in…

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Q: Construct (a) slope field for the DE: y'+4 y'=x (b) tabulate the values at these 5 points ( 0,0 )(…

A: We have to sketch the slope field for the differential equation y'=x2-4y2. Then we  will tabulate…

Q: 5. Compute the lenght of the curve u = t, v=0 from t = 0 to t = 1 using the first fundamental form.

A: Suppose, a continuous function is provided over a certain interval. The arc length is the length…

Q: Find the quadratic approximation at (0,0) for f(x,y) = ln(1+x^2 + y^2).

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Q: estigate whether the function u(x, y, z) = e^(3x+4y)cos(5z) is a harmonic function, where harmonic…

A: ux,y,z=e3x+4ycos5z

Q: Let F = (t + 4) i +² sin(41) j+ t^k, G = e' i+ 4f* cos(t) j+ e$t* k and 512 -i+ 4² j+te²ª k 1 +t H =…

A: Given :  F=(t+4)i+t^ 2 sin(4t)j+t^ 4 k G=e^ t i+4t^ 4 cos(t)j+e^ 5t^ 2  H = (t ^ 5)/(1 + t) * i + 4t…

Q: the

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Q: Consider the following function: H(x, y) 7x'y +-21x'y Find the linear approximation of H(x, y)…

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Q: = (u² - cos v, u + sint, cos v) x(u, v):

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Q: Evaluate the path integral SF. dr, where F(x, y) = yi + xj and C is the line segment from (0, 0) to…

A: The given problem is to evaluate the given path integral of the vector field in the given path of…

Q: 5. Find all analytic functions f such that Ref + Im f = ry.

A: We need to find the analytic functions f such that Re f+Im f=xy. Consider the function of the form…

Q: Find the flux of ♬ through the closed cylinder of radius 3, centered on the z-axis, with 3 ≤ z ≤ 5…

A: