(1 point) Let W(s, t) = F(u(s, t), v(s, t)) where u(1,0) = 1, u,(1,0) =...
(1 point) Let Compute (4,4) (4,4) (1 point) Let W(s,t) - F(u(s, t), v(s, t)) where u(1,05, u,(1,0-7, ua(1,0) 2 F -5,-2)-7,F (-5,-2)4
Let W(s, t) - F(u(s, t), vis, t)), where F, u, and v are differentiable, and the following applies. u(6, -6) - 7 v(6, -6) -9 us(6, -6) - 2 vs(6, -6) -7 (6,-6) --4 V:(6, -6) = 3 Fu(7.-9) - - 1 F (7.-9) - -2 Find W (6, -6) and W.(6, -6). Ws(6, -6) W:(6, -6) =
Question 19: Linear Transformations Let S = {(u, v): 0 <u<1,0 <v<1} be the unit square and let RCR be the parallelogram with vertices (0,0), (2, 2), (3,-1), (5,1). a. Find a linear transformation T:R2 + R2 such that T(S) = R and T(1,0) = (2, 2). What is T(0, 1)? T(0,1): 2= y= b. Use the change of variables theorem to fill in the appropriate information: 1(4,)dA= S. ° Sºf(T(u, v)|Jac(T)| dudv JA JO A= c. If f(x, y)...
Let H=F(x,y) and x=g(s,t), y=k(s,t) be differentiable functions. Now suppose that g(1,0)=8, k(1,0)=4, gs(1,0)=8, gt(1,0)=2, ks(1,0)=1, kt(1,0)=5, F(1,0)=9, F(8,4)=3, Fx(1,0)=13, Fy(1,0)=7, Fx(8,4)=9, Fy(8,4)=2. Find Hs(1,0), that is, the partial derivative of H with respect to s, evaluated at s=1 and t=0.
(1 point) 5x2 — 5у, v %3D 4х + Зу, f(u, U) sin u cos v,u = Let z = = and put g(x, y) = (u(x, y), v(x, y). The derivative matrix D(f ° g)(x, y) (Leaving your answer in terms of u, v, x, y ) (1 point) Evaluate d r(g(t)) using the Chain Rule: r() %3D (ё. e*, -9), g(0) 3t 6 = rg() = dt g(u, v, w) and u(r, s), v(r, s), w(r, s). How...
Let U,V,W be vector spaces over field F, and let S ∈ L(U,V) andT ∈ L(V,W). (a) Show that if T ◦ S is injective, then S is injective (b) Give an example showing that if T ◦ S is injective then T need not be injective. (c) Show that if T ◦ S is surjective, then T is surjective. (d) Give an example showing that if T ◦ S is injective then S need not be surjective.
(1 point) Let S be the surface defined by ř(u, u)-< ucosu, u sinu, u > for (u,t) in D-((mu) : 0 < u < 3,0 < u < π} Evaluate the surface integral of F-<,z,y>upward across S. F-dS = (1 point) Let S be the surface defined by ř(u, u)- for (u,t) in D-((mu) : 0
QUESTION 8 Let V = U ㊥ W where V is a finite-dimensional vector space over a field F, and U and w are subspaces of V. Suppose U1 and U2 are subspaces of U and Wi and W2 are subspaces of W Show that QUESTION 8 Let V = U ㊥ W where V is a finite-dimensional vector space over a field F, and U and w are subspaces of V. Suppose U1 and U2 are subspaces of U...
Question 6:(1 point) Let u (-2, 0, 1), v = (-1,-1, s) and w = (0,-2, t). Find the condition on s and t which makes the set(u, v, w} linearly dependent. For example, if your condition is 2s + 3t + 1 = 0, you would write it in the format 2's+3*t =-1, with s and t on the left hand side and the constant on the right-hand side.
Problem 6. Let V, W, and U be finite-dimensional vector spaces, and let T : V → W and S : W → U be linear transformations (a) Prove that if B-(Un . . . , v. . . . ,6) is a basis of V such that Bo-(Un .. . ,%) s a basis of ker(T) then (T(Fk+), , T(n)) is a basis of im(T) (b) Prove that if (w!, . . . ,u-, υ, . . . ,i)...