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.If T is a mixed tensor, then T} + T} is generally not a tensor a tensor May or may notTensor 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.
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- If vector A = [4/1] and vector B = [-3/2] find A-BSolve for third and fourth item (Vector Calculus)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 D3. (a) (5 marks) Find a vector v such that ||v|| = 5, and v is in the opposite directionof -5i +7j. (b) (5 marks) Let u = 3i - 5j, v = -2i + 7j, Show that (u +v) (u - v) =||u||^2- ||v||^2