Consider F any invertible tensor and Q any orthogonal tensor. (a) If you V-number is odd, show that (QF)-T = QF-T (b) If your V-number is even, show that (FQ)-T = F-TQ
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- In differential geometry give me example of covariant tensor and contra variant tensor.Vector u has POSITIVE x-component, POSITIVE y-component and ZERO z-component. Vector v has NEGATIVE x-component, POSITIVE y-component and POSITIVE z-component. What is true about u x v? There is not enough information to know anything. O It has NEGATIVE x-component, NEGATIVE y-component, NEGATIVE z-component O It has POSITIVE×-component, NEGATIVE y-component, NEGATIVE z-component It has POSITIVE x-component, NEGATIVE y-component, POSITIVE z-componentTensor Auv is type (2 0) and tensor Wv is type (0 1). prove that AuvWv is a tensor of type (1 0) using the contraction property and justify why.
- If T is a mixed tensor, then T} + T} is generally not a tensor a tensor May or may not2.Vector analysis (calculus 4)Calculate the velocity and acceleration vectors and the speed of r(t) = (6+²₁² ²1 1 > at the time t = 4. 6+1² (Use symbolic notation and fractions where needed. Give your answer in the vector form.) v(4) = a(4) = Calculate the speed of r(t) at the time t = 4. (Use symbolic notation and fractions where needed.) v(4) =
- If A and B are both second-order tensors and t, u, v, and w are all vectors, prove the following identities using the indicial notation: (1) A:B= Tr(A·B" ) (2) (txu)·(v×w) = (t - v)(u · w)-(t·w)(u· v) (3) [t·(ux v)]w= (t w)(ux v)+ (u · w)(v×t)+(v·w)(txu)Explain “ some combination gives the zero vector, other than the trivial combination with every x=0.”Format: Find the directional derivative of f(x, y, z) = -4x²z - 11 xy + 2z² at P = (-3, -3, -2) in the direction of (6,-2,0). D(6,-2,0)| (-3,-3,-2) O Search • If your answer below is a vector, input it as a column vector using the vector/matrix palette tool. • Use to explicitly indicate multiplication between variables and other variables, between variables and brackets, or between sets of brackets (e.g. x*y rather than xy; 2*(x+y) rather than 2(x+y)). . Put arguments of functions in brackets (e.g. sin(x) rather than sin x). . Give your answers in exact form. ab sin (a) Əə dx What is the maximum rate of increase of f at P? |Vƒl(-3,-3,-2) sin (a) Ә 2 f W ∞ f 8 a Ω a Ω Submit Assignment N Quit & Save [x] Back < 0 Question M 32 D
