1.5. Show that in the (r, )-space Rn+l the planes (1.3) are characteristic with respect to the wa...
1.1 Be f: R->R given by , Show that f ist convex 1.2 Be f: given by . Show that f ist convex 1.3 Show, that for all applies : f(x) (0.00) → R Oo f(r) =-In(2) We were unable to transcribe this imageInla f(x) (0.00) → R Oo f(r) =-In(2) Inla
A plane electromagnetic wave of circular frequency propagates in free space in the direction of the unit vector . Setting the wave number , show that Note: comes from , and from Faraday's Law, etc. 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 imageWe were unable to transcribe this imageWe were unable...
Please show all work Consider a plane wave -ik.r E(r) = Ege = Eye-i(k+x+kyy+kzz) (1.1) where Eo = amplitude vector constant in space (complex phasor with direction) = E.(w) = Ey(w)|eibe (1.2) k= Ýk +ý k+2 kg r = x + ŷ y + 2 z k(w) = k} + k + k Wype = (1.3) (1.4) 27 (1.5) (a) Using the fact that V E(r) = 0 in the absence of sources, and the vector identity V.(VA) = V(VA)...
Metric: slopes of light in u,r plane: or Question : In the u,r plane, show that r=R is a horizon with opposite characteristics. i.e. light rays starting at r > R cannot enter the r < R region. (This is because of the accelerating expansion of the universe) We were unable to transcribe this imageCOnstant We were unable to transcribe this image COnstant
Consider a second-order linear homogeneous equation Suppose that are two solutions. Show that is also a solution to the equation (plug it in and use the fact that and are solutions). We were unable to transcribe this imageWe were unable to transcribe this imageZhg + th = Eh We were unable to transcribe this imageWe were unable to transcribe this image
Let C be a curve of length L in space and a vector field of constant norm and tangent to C at each point of the curve. What is the work done by along C? Justify your answer. We were unable to transcribe this imageWe were unable to transcribe this image
a) By direct substitution determine which of the following functions satisfy the wave equation. 1. g(x, t) = Acos(kx − t) where A, k, are positive constants. 2. h(x, t) = Ae where A, k, are positive constants. 3. p(x, t) = Asinh(kx − t) where A, k, are positive constants. 4. q(x, t) = Ae where A, a, are positive constants. 5. An arbitrary function: f(x, t) = f(kx−t) where k and are positive constants. (Hint: Be careful with...
Q3) An incident wave travels in free-space along the z-direction and impinges upon a cube of dielectric material with a constant permittivity of ε. The faces of the dielectric cube are parallel to the x-y. y-z, x-z planes. The length of each side of the cube is d. The polarization of the incident electric field is along the x-axis. Use the Born approximation to calculate the scattered electric and magnetic fields in the far-field zone along the ks-R, =+x and...
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...
Let T : C([0, 1]) → R be a (not necessarily bounded) linear functional. Show that T is positive if and only if = (here 1 denotes the constant function [0, 1] → R, x → 1). We were unable to transcribe this imageWe were unable to transcribe this image