Using Green's theorem, evaluate F(r) • dr counterclockwise around the boundary curve C of the region R, where C F = |8y, 2x – 2y*|, R the square with vertices + (4, 4),±(4,–4). F(r) ● dr =

Q: Let 2C R" be open with smooth boundary. Show that (a) Green's function G(r, y), z,yE n, is uniquely…

A: Given:

Q: (a) Let R be a simply connected region and let f : R → C be a holomorphic functio Fix a point z in R…

A:

Q: Show that the function a Y1 (x, y) = [21 2] в с Y2 is an inner product on R? if and only if b2 – ac…

A:

Q: (B) Prove that the stream line of b are counterclockwise circles, for that calculate F = Vx (-y ěx+…

A:

Q: attracting, repelling, or neutral. For racting basin.

A: Fixed point existence: Assume that the function f(z) has a fixed point at z0. Then z0 is as…

Q: Let F(t) = (- t + 1, t – 2t", - 2t). Find the line (L) tangent to r (t) at the point (- 15, – 24, –…

A:

Q: Let fj= f(xj), f = f'(xj), ƒ!! = f'(x;) and so on, where ; are equispaced grid points with spacing…

A: Solution:

Q: Use Green's Theorem to evaluate Se (y? – tan-1 (r)) dr + (3r + sin (y)) dy, where C is the boundary…

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

Q: Let : C→C be the function defined by ƒ(z) = z and let C be the closed anticlockwise contour whose…

A:

Q: let T: P4(R)→R2 be the map given by T(f)=(f(1),f"'(1)) where P4(R) is the set of polynomials of…

A:

Q: (a) Evaluate f. (sin x + 3y²)dx + (2x – e-y*)dy, where C is the boundary of the half-disk x² + y? <…

A:

Q: (5) Let D = {(x, y) E R? ||r| + |y| < 1}. The region D is a square diamond () in the ry-plane with…

A:

Q: Let C be the closed curve oriented in the counterclockwise direction, consisting of the line segment…

A:

Q: f(x) is a natural cubic spline function having knots {-2, 0, 1), defined by ƒ(x) = { g(x), arª+x³+…

A:

Q: Consider the mapping w = e^−iz of the domain D = { z belongs to C : Im(z) > 0 , −π/2 < Re(z) < 0 }…

A:

Q: Let f(z) = E (z – 5)" n4" (n + 2)2 n=1 i) ii) iii) Find the region where the function f(2) is…

A: The boarder of the image looks like square root of the term. But not it is square root. Please check…

Q: Let F(x, y, z) = (x-1z, y¬lz, In(xy). (a) Verify that F = V ƒ, where ƒ(x, y, z) = z In(xy). (b)…

A: Given Fx, y, z=x-1z, y-1z, lnxy (a) Verify for F=∇f where fx, y, z=z lnxy The gradient of a function…

Q: use the inner product = p(x)q(x)dx. to find the projection of p(x) q(x) = 5x + 2. = S'; x3 onto

A:

Q: Evaluate S. (2xy – x²)dx + (x + y²) dy such that c is a curve bounded by y? – x , y – x² ?

A:

Q: (a) Use a Green theorem to evaluate (y¹ - 2y)dx - (6x - 4xy³). dy where c is the boundary of the…

A:

Q: The absolute maximum value of ƒ(x, y) = 4xy² — x²y² – xy³ on the triangular region in the xy-plane…

A: Given that, f(x,y) = 4xy2 - x2y2 - xy3 and a triangular  region D with vertices (0,0) ,(6,0) and…

Q: Let, T: R3 → R³;T(x, y, z) = (2x + y, y – z,2y + 4z) Find the rank and nullity of T.

A:

Q: Let Ci be the rectangle with vertices (0,0), (2,0), (0,1), and (2,1) oriented counterclockwise, and…

A: Let C₁ be the rectangle with vertices (0,0). (2,0), (0, 1), and (2,1) oriented counterclockwise, and…

Q: f: R\{0} → R, f(x) = +5z-2-1 e

A:

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.