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7. Consider an N-dimensional rectangular space. Show that the divergence of the position vector in this...
Consider the following region R and the vector field F. a. Compute the two-dimensional divergence of...
Consider the following region R and the vector field F. a. Compute the two-dimensional divergence of the vector field. b. Evaluate both integrals in the flux form of Green's Theorem and check for consistency. c. State whether the vector field is source free. F = (8xy,9x2 - 4y?); R is the region bounded by y = x(6 - x) and y=0. .- a. The two-dimensional divergence is 0 b. Set up the integral over the region. dy dx 0 Set...
QUESTION 1 A quantity, 7", in an n-dimensional space that has n values and transforms between...
QUESTION 1 A quantity, 7", in an n-dimensional space that has n values and transforms between reference frames S and S in the same way as an infinitesimal displacement (dx), that is: is called a contravariant vector Let T be a contravariant vector in a 2-dimensional space, and let S be defined in Sas: 4 and 12 with A = 3 and B = -2 Find the second component, 7", of the vector in S, at point P, where the...
Show work 1. Assume you are given an mxn matrix H, an n-dimensional vector spacev h:VW...
Show work
1. Assume you are given an mxn matrix H, an n-dimensional vector spacev h:VW are there such that H RepBp(h)? B, and an m-dimensional vector space Wwith basis D. How many linear mappings A) None B) One C) It depends on m and n. D) Infinitely many 2. Which of the following matrices will change from the basi the basis(31),(7))? R2 to -1'1 3 -2 B) 3 2 A) C) D) 3. Assume you are given an n-dimensional...
For Problems C4-C11, prove or disprove the statement. C4 If V is an n-dimensional vector space...
For Problems C4-C11, prove or disprove the statement. C4 If V is an n-dimensional vector space and {11,...,Vk} is a linearly independent set in V, then k sn. C5 Every basis for P2(R) has exactly two vectors in it. C6 If {V1, V2} is a basis for a 2-dimensional vector space V, then {ağı + bū2, cũı + dv2} is also a basis for V for any non-zero real numbers a,b,c,d.
Consider the following region and the vector field F. a. Compute the two-dimensional divergence of the...
Consider the following region and the vector field F. a. Compute the two-dimensional divergence of the vector field. b. Evaluate both integrals in Green's Theorem and check for consistency F= (2x-2y); R=(x,y): x2 + y²59 a. The two-dimensional divergence is (Type an exact answer.) b. Set up the integral over the region. Write the integral using polar coordinates with r as the radius and O as the angle SO rdr d0 (Type exact answers.) 0 o Set up the line...
9.4 (Idempotent endomorphisms) Let V be an n-dimensional vector space over Kand let End(V) such t...
9.4 (Idempotent endomorphisms) Let V be an n-dimensional vector space over Kand let End(V) such that f-f. a) Show that fis diagonalizable and that V-rge f b) Determine x and . f ker f.
9.4 (Idempotent endomorphisms) Let V be an n-dimensional vector space over Kand let End(V) such that f-f. a) Show that fis diagonalizable and that V-rge f b) Determine x and . f ker f.
Problem 5 Let U be an n dimensional vector space and T E L(U,U). Let I...
Problem 5 Let U be an n dimensional vector space and T E L(U,U). Let I denote the identity transformation I(u) = u for each u EU and let 0 denote the zero transformation. Show that there is a natural number N, and constants C1, ..., CN+1 such that C1I + c2T + ... + CN+1TN = 0 (Hint: Given dim(U) = n, what is the dimension of L(U,U)? consider ciI + c2T + ... + Cn+11'" = 0, where...
2. Consider a three-dimensional Universe. A vector of this space, starts from the origin of the...
2. Consider a three-dimensional Universe. A vector of this space, starts from the origin of the coordinate system and has the tip described by the coordinates 1, 0, a) Write the matrix that describes a rotation of this three dimensional vector about the Oz axis by an angle of 45° Both the initial and the final coordinates have the same origin. b) Calculate the projections (of the tip) of this vector along the new axes of coordinates.
Consider a two-dimensional state vector space and a basis in this space lay), laz), eigenvectors of...
Consider a two-dimensional state vector space and a basis in this space lay), laz), eigenvectors of an observable A: A|az) = ajlaj) Alaz) = azlaz) A representation of Hamiltonian operator in this basis is: h = (9) - Calculate the dispersion of measurements of A and Hat time t. Is the uncertainty principle Energy x Time fulfilled?
7 Harmonic oscillator in "energy space" Consider the harmonic oscillator in "energy space", i.e., in terms...
7 Harmonic oscillator in "energy space" Consider the harmonic oscillator in "energy space", i.e., in terms of the basis of eigenvectors n) of the harmonic oscillator Hamiltonian, with Hn) -hwn1/2)]n). We computed these in terms of wavefunctions in position space, ie. pn(x)-(zln), but we can also work purely in terms of the abstract energy eigenvectors in Dirac notation. PS9.pdf 1. You computed the matrix elements 〈nleln) on an earlier problem set. Now find (nn) for general n,n' 2. Find the...
Consider the following region R and the vector field F. a. Compute the two-dimensional divergence of the vector field. b. Evaluate both integrals in the flux form of Green's Theorem and check for consistency. c. State whether the vector field is source free. F = (8xy,9x2 - 4y?); R is the region bounded by y = x(6 - x) and y=0. .- a. The two-dimensional divergence is 0 b. Set up the integral over the region. dy dx 0 Set...
QUESTION 1 A quantity, 7", in an n-dimensional space that has n values and transforms between reference frames S and S in the same way as an infinitesimal displacement (dx), that is: is called a contravariant vector Let T be a contravariant vector in a 2-dimensional space, and let S be defined in Sas: 4 and 12 with A = 3 and B = -2 Find the second component, 7", of the vector in S, at point P, where the...
Show work
1. Assume you are given an mxn matrix H, an n-dimensional vector spacev h:VW are there such that H RepBp(h)? B, and an m-dimensional vector space Wwith basis D. How many linear mappings A) None B) One C) It depends on m and n. D) Infinitely many 2. Which of the following matrices will change from the basi the basis(31),(7))? R2 to -1'1 3 -2 B) 3 2 A) C) D) 3. Assume you are given an n-dimensional...
For Problems C4-C11, prove or disprove the statement. C4 If V is an n-dimensional vector space and {11,...,Vk} is a linearly independent set in V, then k sn. C5 Every basis for P2(R) has exactly two vectors in it. C6 If {V1, V2} is a basis for a 2-dimensional vector space V, then {ağı + bū2, cũı + dv2} is also a basis for V for any non-zero real numbers a,b,c,d.
Consider the following region and the vector field F. a. Compute the two-dimensional divergence of the vector field. b. Evaluate both integrals in Green's Theorem and check for consistency F= (2x-2y); R=(x,y): x2 + y²59 a. The two-dimensional divergence is (Type an exact answer.) b. Set up the integral over the region. Write the integral using polar coordinates with r as the radius and O as the angle SO rdr d0 (Type exact answers.) 0 o Set up the line...
9.4 (Idempotent endomorphisms) Let V be an n-dimensional vector space over Kand let End(V) such that f-f. a) Show that fis diagonalizable and that V-rge f b) Determine x and . f ker f.
9.4 (Idempotent endomorphisms) Let V be an n-dimensional vector space over Kand let End(V) such that f-f. a) Show that fis diagonalizable and that V-rge f b) Determine x and . f ker f.
Problem 5 Let U be an n dimensional vector space and T E L(U,U). Let I denote the identity transformation I(u) = u for each u EU and let 0 denote the zero transformation. Show that there is a natural number N, and constants C1, ..., CN+1 such that C1I + c2T + ... + CN+1TN = 0 (Hint: Given dim(U) = n, what is the dimension of L(U,U)? consider ciI + c2T + ... + Cn+11'" = 0, where...
2. Consider a three-dimensional Universe. A vector of this space, starts from the origin of the coordinate system and has the tip described by the coordinates 1, 0, a) Write the matrix that describes a rotation of this three dimensional vector about the Oz axis by an angle of 45° Both the initial and the final coordinates have the same origin. b) Calculate the projections (of the tip) of this vector along the new axes of coordinates.
Consider a two-dimensional state vector space and a basis in this space lay), laz), eigenvectors of an observable A: A|az) = ajlaj) Alaz) = azlaz) A representation of Hamiltonian operator in this basis is: h = (9) - Calculate the dispersion of measurements of A and Hat time t. Is the uncertainty principle Energy x Time fulfilled?
7 Harmonic oscillator in "energy space" Consider the harmonic oscillator in "energy space", i.e., in terms of the basis of eigenvectors n) of the harmonic oscillator Hamiltonian, with Hn) -hwn1/2)]n). We computed these in terms of wavefunctions in position space, ie. pn(x)-(zln), but we can also work purely in terms of the abstract energy eigenvectors in Dirac notation. PS9.pdf 1. You computed the matrix elements 〈nleln) on an earlier problem set. Now find (nn) for general n,n' 2. Find the...
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