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What are covariant and contravariant vectors? Show that acceleration is a contravariant vector.

Q. A second order tensor has components ■(2&1&0@1&-3&-1@0&-1&2) in OX1X2X3. The transformation between the coordinates X1, X2, X3 and X1’ , X2’ , X3’ is defined by

X1’=1/3 (2x1+2x2-x3)

X2’=1/3 (2x1-x2+2x3)

X3’=1/3(-x1+2x2+2x3)

(i)Find its components in the new system OX1’X2’X3’.(By using Matrix Formula).

(ii) Compute A’32 in the new system.(By using component formula)

X1’=1/3 (2x1+2x2-x3)

X2’=1/3 (2x1-x2+2x3)

X3’=1/3(-x1+2x2+2x3)

(i)Find its components in the new system OX1’X2’X3’.(By using Matrix Formula).

(ii) Compute A’32 in the new system.(By using component formula)

Q. Write transformation equations for a second order tensor Aij w.r.t this transformation and verify A’ii=Aii

WHAT IS MEAN BY (0,2) and (2,0) (1,1) tensor.

For the following matrix calculate the determinant:

{{5,3,7},{1,0,1},{9,6,4}}

{{5,3,7},{1,0,1},{9,6,4}}

Evaluate the following:

{{1,2,3},{4,5,6},{7,8,9}} * {{7,4},{6,4},{5,3}}

{{1,2,3},{4,5,6},{7,8,9}} * {{7,4},{6,4},{5,3}}

Given the following:

A = 3 * 3

B = 2 * 4

C = 4 * 3

D = 3 * 2

Which of the following are defined:

A * D

D * B

A * D^T

C * D + A * D^T

A * B

A * D^T + C

A = 3 * 3

B = 2 * 4

C = 4 * 3

D = 3 * 2

Which of the following are defined:

A * D

D * B

A * D^T

C * D + A * D^T

A * B

A * D^T + C

Evaluate the following problem:

2*{{3,5},{7,9}} + 3*{{7,6},{5,4}} - 4*{{9,8},{7,6}}

2*{{3,5},{7,9}} + 3*{{7,6},{5,4}} - 4*{{9,8},{7,6}}

Solve the system of simultaneous equations below using any of the matrix method

2x + 3y - z = 6

3x + y + 5z =8

x + 2y + z =10

2x + 3y - z = 6

3x + y + 5z =8

x + 2y + z =10

4x + 2y + 3z. = 35

x + 3y + 2z. = 45

2x + y + 5z. = 28

Calculat value of x,y & z

Crsmer's rule.

x + 3y + 2z. = 45

2x + y + 5z. = 28

Calculat value of x,y & z

Crsmer's rule.