The main point of this exercise is to use Green’s Theorem to deduce a special case of the change of variable formula. Let U, V ⊆ R2 be path connected open sets and let G : U → V be one-to-one and C2 such that the derivate DG(u) is invertible for all u ∈ U. Let T ⊆ U be a regular region with piecewise smooth boundary, and let S = G(T). Solve C D E

The main point of this exercise is to use Green’s Theorem to deduce a special case of the change of variable formula. Let U, V ⊆ R2 be path connected open sets and let G : U → V be one-to-one and C2 such that the derivate DG(u) is invertible for all u ∈ U. Let T ⊆ U be a regular region with piecewise smooth boundary, and let S = G(T). Solve C D E

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

Find a Schwarz-Christoffel transformation of the upper half-plane (UHP) onto the subset of the UHP bordered by the positive real axis, the vertical segment {iy: 0 0 is a positive constant, and the infinite horizontal segment H = {x + Li: x <0}. V
Let the linear map f:R³-R2 be defined by f(x,y,z) = (x-y+z, x+2z). Then the matrix associated with f relative to the standard bases of R³ and R² is: OA 1 02)
7. Let V be the vector space of polynomials in two variables x and y of degree at most two: V = {ax² + bxy + cy² + dx + ey + ƒ | a, b, c, d, e, ƒ € R}. Let T be the linear operator on V defined by T(g(x, y)) = Ə Ix 9 (x, y) + Find the Jordan canonical form of T. Ə dy I (x, y). ду
Q3. Let L: R2-R2 be a mapping from the vector space R2 to itself defined by L(x , y)= ( 2x+y, x-3y) for each (x,y) belong to R?. Show that L is a linear operator on R² and then find the representation matrix ms(L), where S is the standard bases of R2.
Let V be a finite-dimensional inner product space over C with inner product (, ), and let a, b e V \0. Define T: V + V by T(v) = (v, a) b for all v E V. (i) Show that T is a linear map. (ii) For v E V, find T*(v) in terms of v, a, b. (iii) Prove that if T = T*, then b = Xa for some A E R.
Find a) the kernel, b) the image, c) the null and d) the rank of the transformation linear x + z T: R* → R²; T( % ) = Giw) y y+w
(3) Suppose that T: V →F is a linear map. Prove that if u E V such that u 4 Ker(T) then V = Ker(T) {au | a E F}.
4. Let T : R5 → R³ be a surjective linear map. What can you say about the dimension of ker(T) ?
Let f : R? → R be defined by f((x, y)) = 5x + 7y. Is f a linear transformation? a. f({x1,Yı) + (x2, Y2)) : f({x1, Y1)) + f({x2, Y2)) = + Does f((x1,Y1) + (x2, Y2)) = f(x1, Y1)) + f((x2, Y2)) for all (01, Y1), (x2, Y2) E R?
a) let T : R3 → R be a linear transformation defined by T(a, b, c) = 2a−b+ 5c. Find y ∈ R3such that T(x) =< x, y > for all x ∈ R3.b) Let T : V → V be a linear operator over a finite dimensional vector space V . Prove thatdim(ker(T)) + dim(ImmT) = Dim(V )
Let a1 = col [1, 1,0] and r2 = col [1, 1, 1]. (a) Apply GSOP to (1,x2) to generate an orthonormal set ( v1, v2 ). %3D %3D (b) Find orthogonal projections of x span (1,22 ). col [0, 1, 1] on S1 = span (r1 ) and on S2