Suppose that for each choice of a contravariant vector (a vector) , the quantities are defined at each point through a linear relationship of the form
transform like a covariant vector (1-form). Prove that the quantities transform like a tensor of type (0,2) at each point.
Suppose that for each choice of a contravariant vector (a vector) , the quantities are define...
Suppose that for each choice of a contravariant vector (a vector) , the quantities are defined at each point through a linear relationship of the form transform like a covariant vector (1-form). Prove that the quantities transform like a tensor of type (0,2) at each point. A" (r) B,(z) We were unable to transcribe this imageWe were unable to transcribe this image A" (r) B,(z)
We can combine the scalar potential V and the vector potential A to a combined 4-vector potential: Calculate the components of a 4x4 electromagnetic field tensor: with the contravariant vector: from the electric field and the magnetic field We were unable to transcribe this imageWe were unable to transcribe this imageい() ct OA Ot We were unable to transcribe this image い() ct OA Ot
Using FTLM. a) Let . Use linear algebra to prove that there is a polynomial such that p + p' - 3p'' = q. Hint: consider the map defined by Tp: p + p' - 3p'', and use FTLM. b) Let be distinct elements of . Let be any elements of . Use linear algebra to prove that there is a such that Hint: consider the map defined by . You can use any facts from algebra about the solution...
Show that the correlation matrix of any random vector X is nonnegative definite, where the correlation matrix is defined by , (Assume we know that the covariance matrix of X denoted is defined by is nonnegative definite, and . Re IRmxm We were unable to transcribe this imageVar(Xi)Var(X ат ат We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this...
Suppose is a finite dimensional vector space. For hyperplanes in say they are linearly independent provided the corresponding linear subspaces in are linearly independent. Set and show that are linearly independent if and only if . (Hint: Write for and consider by ). We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageimnH...
A) Find a vector valued function of the form for the paraboloid . B) Find a vector valued function for the elliptic cylinder r(u, v) = x(u, v)i + y(u, v)j + z(u, v)k We were unable to transcribe this imageWe were unable to transcribe this image
Note: In the following, if is a set and both and are positive integers, then matrices with entries from . The problem below has many applications. If is a linear map from complex vector space to itself, and is an eigenvalue of , then is a simple eigenvalue of if . 1. Suppose is a vector space of dimension over field where you may assume that is either or , and let be a linear map from to . Show...
Let V be a finite-dimensional vector space and let T L(V) be an operator. In this problem you show that there is a nonzero polynomial such that p(T) = 0. (a) What is 0 in this context? A polynomial? A linear map? An element of V? (b) Define by . Prove that is a linear map. (c) Prove that if where V is infinite-dimensional and W is finite-dimensional, then S cannot be injective. (d) Use the preceding parts to prove...
Suppose that the vector field, , is continuously differentiable and satisfies in the interior of the domain , open and bounded, whose boundary is a smooth surface (at least class) , steerable. Show that cannot be tangent to in every point of the surface We were unable to transcribe this imagedivF = 0,Fi + OyF2 +0. F3 > 0 Ωε P3 We were unable to transcribe this image11 We were unable to transcribe this imageWe were unable to transcribe this...