Show that if a E C, k E E Z, and y is parametrized by z(t) = a +re“, t E [0, 2], then |(2 - a)* dz = 0 if k + -1, whereas | (z – a)-1 dz = 2n i.

Q: Find z dV, where E is the solid tetrahedron with vertices (0,0,0), (1,0,0), E (0,4,0), and (0,0,2)

A: The vertices of the tetrahedron are (0,0,0),(1,0,0),(0,4,0),(0,0,2).  x varies from x:0→1y varies…

Q: x y 11. Use Green's theorem to evaluate O In(1+ y) dx –- dy , where C is the triangle with vertices…

A:

Q: Express the given parametrization in the form y f(x) by eliminating the parameter. - In(t), y = 5 +…

A:

Q: From the Laplace Equation show that there exist solutions in the form X(x) +Y(y) + Z(z)

A:

Q: Verify that u is a solution to the equation ugx + Uyy + u zz = 0. u(x, y, z) = e4z+3y cos(5z)

A: Follow the procedure given below.

Q: Let C be the line segment from [1 – i,2+ 2i). (a) Parametrize C. (b) Integrate 2 over C.

A: According to the given information, Let C be the line segments from [1-I,2+2i]. For part (b) it is…

Q: Verify Green's Theorem by evaluating both integrals y2 dx + x² dy = dA ду for the given path. C:…

A: In this question we can verify the Green's Theorem . i.e, ∫C M dx +Ndy = ∫∫R ∂N∂x-∂M∂ydA .

Q: 2. If f e R[a, b], show using the properties of Riemann Integrability that there exists M > 0 such…

A:

Q: az əz 1. ax' dy If z = In(e* + e"), show that %3D %3D

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

Q: Two functions u and v of x and y are said to satisfy the Cauchy- Riemann equations if ux = v, and u,…

A: Given u = x2-y2v = 2xy Then ux = 2xuy = -2yvx = 2yvy = 2x

Q: The parametric form of the solutions of the PDES Uy + u u z = 0 , u(x,0) = -x is

A:

Q: 4. Find the work done by the force field F = (x? sin(y),y) on a particle that moves along the cubic…

A: Given: The force field, F→=x2siny,y on a particle that moves along the cubic y=x3 from 1,1 to 2,8.

Q: Find a solution to the Dirichlet boundary value problem for a disk. 2u 1 du 1 Pu = 0, r dr r2 a02 +…

A: Answer is mentioned below

Q: . Solve the following Dirichlet problem on a disk V²U(r,0) = 0, 0<r<1, -asosa U(1,0) = 0, - <o< T.

A: This question based on Dirichlet problem on a disk from partial differential equation.

Q: 2. Let W be the union of the first and third quadrants in the xy- plane. That is, let W = xy 0 a. If…

A: According to our company gardening can answer only one question. If a specific questions are asked…

Q: Evaluate // e² dV where E is enclosed by the paraboloid z = 2 + x² + y², the cylinder E x² + y? = 3,…

A: Since we are given with the multiple number of questions and answered according to the guideline.

Q: Compute N(0) if r(t)= and prove that T(0) and N(0) are orthogonal

A: Given rt=&lt;t sin t+cos t,2t2,sin t -t cos y&gt; Then vector is given by r→t=t sin t+cos…

Q: Exercise 4.3.8 Show that if A = E oiu;vf , then At = Ei=1 0,'v¿u

A: Given function is, A=∑σiuiviTi=1r As we know that, A+B+=A++B+ So, we can write the above equation…

Q: Consider the cune C: (x-8)+ 4-7) When parametneing C, A tungent veotoar is i O a) r (0) = (8 +3…

A:

Q: Solve the Laplace equation subject to the boundary conditions, u,(0, y)= 0, u(2, y) = 0, u, (x, 0) =…

A: The Laplace equation is: ∂2u∂x2+∂2u∂y2=0 The given boundary conditions are: ux0,y=0, ux2,y=0,…

Q: Find a solution to the Dirichlet boundary value problem for a disk. zu 1 du 1 Pu = 0, + -- ar? Osr<…

A:

Q: Show that the general solution to the two-dimensional Laplace's Equation. et = 0 is (Acosh(tx)…

A: We have to show that the general solution to the two-dimensional Laplace's Equation…

Q: { e, a| 1, The Fourier integral of f (x) = %3D A) roo w sin w-cOS w sin zwdw w2 roo sin w-CoS W sin…

A:

Q: Find a piecewise smooth parametrization of the path C. 0 st<1 (ti+tj r(t) = %3D 1<t<2 (1, 1) y=x

A: Given rt=ti + tj                             0≤t≤1                                       1≤t≤2

Q: Express the given parametrization in the form y = f(x) by eliminating the parameter. x = 5t + 2, y =…

A: Given

Q: show that fGx) = C,ex +C,e* s a solution to y for all real conatanto C, +C2, and then find C,, C,…

A:

Q: Find z dV, where E is the solid tetrahedron with vertices (0,0,0), (1,0,0), (0,2,0), and (0,0,4)

A: x varies from 0 to 1&amp; y varies from 0 to 2 and z varies from 0 to 4.

Q: əz * əy? ax2 ar? r ar 32. Show that the Laplacian operator a/Əx² + a²/ay² + a /az? in three…

A:

Q: Suppose that the polar coordinates of a moving particle at time t are (r(t), 0(t)). Prove that the…

A:

Q: Solve the BVP for the disk (exterior Dirchlet problem) v²u = 0, 1< r <∞ u (1,0) = sin 0 + cos 20

A:

Q: (1.) Calculate the proper distance between two free particles, one at the origin of a coor- dinate…

A:

Q: of the b) nd y- Fons c) with point.

A:

Q: 1 Evaluate dV, where E lies between the spheres x2 + y2 + z2 = 9 and x2 + y? + z2 x2 + y? + z2 36 in…

A: we want to find the triple integral ∫∫∫E1x2+y2+z2  dVwhere E lies between the sphere x2+y2+z2=9  and…

Q: Evaluate p(10ey – 9x)dy – (yx – 5 In x)dx by using Green's Theorem where C is a trapezium with…

A:

Q: 2. Let i = y?x + xŷ + z²ż, and let S be the paraboloid z = x2 + y? with 0 < z < 1. Show that fci. de…

A: Given  v→=y2x^+xy^+z2z^ and S be the paraboloid defined by  z=x2+y2      , 0≤z≤1 The boundary C of…

Q: 1. Detanmne the rusutt OF a. Gradi ent f(x.y,z) ex sin cy) In (7) b. Divergenie =7? C Curl F=y2 t…

A:

Q: If w = x²y - y²z³,x = rCost,y = rSint, dw z = re' , (r = 2)and(t = 0)then dt

A: Given equations - w = x2y-y2z3       ...............(1)x = rcost        ............(2)y = rsint…

Q: Example 6: Evaluate ez dz dy dx E Where E is enclosed by the paraboloid Z = 3+ x? + y² , the…

A: The given problem is to evaluate the given triple integral of given function ez for which region E…

Q: Use Green's Theorem to evaluate the line integral (y – x)dx + (2x - y)dy for the given path. C: x =…

A:

Q: The solution of Laplace equation du dx du %3D 0, where 0 < x < 8 and 0 < y < 8, with boundary…

A: The given Laplace equation is, ∂2u∂x2+∂2u∂y2=0 Using the second order finite central difference…

Q: Use Green's Theorem to evaluate x' dx + xy dy where C is the triangle with vertices (0,0), (1,4),…

A:

Q: 2. Repeat the above question with a Hamiltonian of the form H(p, T) : +V(#), 2m where p(t) and (t)…

A: repeat the above question with a hamiltonian of thr form Hp→,x→=p→22m+Vx→ where p→t , x→t are vector…

Q: 8 4 Ĝ(Xx, y, z) = 2x +y 2xz, 2х + у Зу? сos(Зпz),3пуз sin(Злz) — х2 1. Determine the potential…

A: For the solution follow the next steps.

Q: 3.use Greens theorem s (1+x5 dx + 2xy dy ,where C is triangle With vertices (0,0),(1,0),(1,3)…

A: Apply greens theorem

Q: Verify that the function f(r, y) = In (x² + y²) is a solution of the Laplace equation %3D Zzz+żyy =…

A:

Q: Find z dV, where E is the solid tetrahedron with vertices (0,0,0), (5,0,0), (0,6,0), and (0,0,3)

A: This question is related to triple integral, We will solve it using given information.

Q: Solve the BVP for the disk (exterior Dirchlet problem) v'u = 0, 1< r <o∞ u (1,0) = sin 0 + cos 20

A: The given function is ∇2u=0 The given condition is u(1,θ)=sinθ+cos 2θ We have to find the general…

Q: Show that the function u(x, y, z) = e3x+4ysin(5z) satisfies the Laplace equation in R3

A: A function u(x,y,z) is said to satisfy the Laplace equation in ℝ3 if u(x,y,z) satisfies…

Q: Find z dV, where E is the solid tetrahedron with vertices (0,0,0), (2,0,0), (0,4,0), and (0,0,1)

A: We have to find the volume integral of the given function.

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.