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: Find order and degree of y'+ (y")? =(x+y")? O a. 2,1 O b. 1,2 О с. 1,1 O d. 2 ,2

A: The given differential equation is                                              y'+(y'')2=(x+y'')2…

Q: Show that the curve r= Vti + Vtj + (2t – 1)k is tangent to the surface x +y² – z = 1 when t=1

A: Given that r=ti+tj+2t-1k and f=x2+y2-z-1 Take the derivative for f, ∇f=ddx,ddy,ddzf=2x,2y,-1…

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 -…

A:

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 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: Assume that F= (e^x−y)i+ (2y−x)j, C1 is curve from (0,1) to (2,5) along the graph of y=x^2+ 1, and…

A: Given , F=ex-yi^+ (2y-x)j^ C1 is a curve from (0,1) to (2,5) along the parabola y=x2+1 C2 is a line…

Q: 3 z'= 2(x"+Z) Arrang e in linear form: dy Q) or dx

A:

Q: (8) Find the Integrating Factor of the equations and Solve it:- a)2 Coshx Cosydx=Sinhx…

A: Since you have posted a question with multiple sub-parts, we will solve first three sub-parts for…

Q: Find the partial derivative at point(u,v)=(-4,0) of w=ln(xa+yb+za),x=uevsin u y=uevcos u, z=uev,…

A:

Q: tan y dy + In (secy) In (6)sec® de

A: Given data:     The given integral is: . The given integral can be written as, Assume Let…

Q: Describe the bifurcation when A is the parameter.   dy/dt = e^-y^t +A

A: The differential equation you've provided is a first-order ordinary differential equation (ODE) with…

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: 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: 2) Find the work of F = (x² - y² -4,2xy) on the semi-circle of radius 2 centered at the origin…

A: ∮F·dr=∮x2-y2-4dx+2xydy             =∬S∂∂x2xy-∂∂yx2-y2-4 dxdy               By applying Greens…

Q: P = V^2*R/ (R+r)^2' V and R are costant r is a variable  what is (dP)/(dr)=

A: Given  P=V2R(R+r)2 Here V and R are constants  r is a variable Find dPdr

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

A:

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: OLif f(x, y,z)=2exp(-5xyz)cos(15z) Find

A: Partial derivatives

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

A: Given,          fx,y,z=eyz+sinx

Q: Let f(x, y, z) = x^2 + y^3 + z^5 and let u =. Find the directional derivative Du f(x, y, z).

A: The directional derivative Du​f(x,y,z) is 64/81​.Explanation:Step 1: To find the directional…

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: 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: the

A:

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

A:

Q: Find the length of the curve 2 ܐܨ 1, a² 6² from the point (a, 0, 0) to the point (x, y, z). x = a…

A: The equation of the given curve is x2a2-y2b2=1, x=acoshza To find the length of the curve from the…

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…