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 defined a...
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) ь,(x ) We were unable to transcribe this imageWe were unable to transcribe this image A" (r) ь,(x )
5(10pts) Consider a general co-ordinate transformation; How does a contravarint vectorV Transform A covariant vector V what is the metric ternsor, gik,how is it defined; How do you obtain, g" From gik How can you use the metric tensor to go from a contravariant vector component to a covariant component gk for a spherical surface of a sphere of radius, R Find guk, and 5(10pts) Consider a general co-ordinate transformation; How does a contravarint vectorV Transform A covariant vector V...
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...
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
Let V be a Hilbert space. Let S1 and S2 be two hyperplanes in V defined by Let be given. We consider the projection of y onto , i.e., the solution of (1) (a) Prove that is a plane, i.e., if , then for any . (b) Prove that z is a solution of (1) if and only if and (2) (c) Find an explicit solution of (1). ( d) Prove the solution found in part (c) is unique. We...
a) The following vector field State whether the divergence of at point A is positive, negative or zero. b) Say if the rotational of at point B is a null vector, which points in the direction of the z-axis or points in the negative direction of z. We were unable to transcribe this image履 2 0 2 4 We were unable to transcribe this imageWe were unable to transcribe this image 履 2 0 2 4
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...
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...