Test: Mathematical Physics - 1

The projection of vector   on vector 

Find a_n of a Fourier series for |x|, -π < x < π

Kroncker delta Sⁱ_j is a mixed tensor of rank _____

Real part of the  is ______ (upto two decimal places)

Given z³ = 1. Let z₀,z₁ and z₂ be the complex roots of the above equation.If z₀ = 1, then the value of z₁z₂ is ____ (Answer should be an integer)

Find the value of integral,  is ____ (upto one decimal place)

The dimensionality  of the vector space of hermitian 3 x 3 matrices is ____ (answer should be an integer)

Given vector  the line integral  where C is a circle of radius 5 units with its center at origin is ________

The determinant of the metric tensor corresponding to ds² = 5(dx¹)² + 3(dx²)² + 4(dx³)² - 6dx¹dx² + 4dx²dx³ is

Consider a vector v = (v₁, v₂, v₃) in three dimensional complex vector space c³. A linear operator T is designed as follows
T( v₁, v_2, v₃) = ( v_1, v₂ - v_3,iv₂)
Find T⁺ matrix representation using orthonormal basis

Given the Legendre polynomial P₀(x) = 1, P₁ (x) = x and  then polynomial (3x² + x -1)

The matrix A defined by  is orthogonal if

Find the inverse Laplace transform of f(s) = 

Find the complex coefficient Cn of the fourier series of the function  for n is odd.

The equation of the plane that is tangent to the surface xyz = 8 at point (1,2,4) is

The value of the integral  is ______ (upto two decimal places)

The value of the Contour integral
 
and the contour C is a circle of radius 2 centred at the origin traversed in the counterclockwise direction is ______ (answer should be an integer).