What are symmetric and antisymmetric tensors? Show that the dot product of two
3-D vectors is a scalar.
A tensor Tijk...is said to be symmetric in a pair of indices (say ,j) if
or anti-symmetric in i,jif
For a second rank tensor we need not specify the indices as there are only two to choose from! For example,δ ij is symmetric; ijk is anti-symmetric in any pair of indices.
Note: if Aij is a symmetric second rank tensor then the matrix corresponding to
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Using Rodrigue’s formula, obtain the expression for P3 ( x) .
A metal plate covering the first quadrant of the xy plane has its edge along the yaxis
kept at T = 0, and the edge along the x-axis is kept at an initial-temperature:
Obtain the steady state temperature distribution as a function of x and y.
Show that
i) d/dx [x m J m( x)]= x m Jm−1( x)
- ∫ xJ0 (λx)dx = 1/ λ J1(λ)
iii) ∫ x e− x H1( x) H 2 ( x)dx = 4sqrt( π)
Determine the value of x, for which the matrix, P
is singular.
Obtain the Laurent series expansion of the function ez/(z -1)2 about z = 1, and determine the type of singularity.
Using the method of Laplace transform, solve the following initial value problem:
y′′ − 2 y′ + y = et ; y(0) = −2, y′(0) = −3
What are symmetric and antisymmetric tensors? Show that the dot product of two
3-D vectors is a scalar.