Mathematical Physics
Mathematical Physics in GATE PH tests the analytic machinery used across physics: complex analysis, linear algebra, differential equations, vector calculus, transform methods, and probability. Questions typically combine a standard technique with a physical boundary condition, so most marks are lost while setting up the problem rather than during the final algebra.
Core rules and tools
- A complex function is analytic where the Cauchy-Riemann equations hold and the partial derivatives are continuous. Poles determine residues; the residue theorem converts a closed contour integral into 2πi times the enclosed residues.
- Hermitian matrices have real eigenvalues and orthogonal eigenvectors. For real symmetric matrices, diagonalisation is always possible; degeneracy requires care when forming a basis.
- Second-order linear ODEs are classified by the indicial equation at a regular singular point. Sturm-Liouville problems give orthogonal eigenfunctions with real eigenvalues.
- Vector calculus identities such as ∇·(∇×A)=0 and ∇×(∇φ)=0 are frequent distractors. The divergence theorem and Stokes' theorem require the correct orientation and boundary.
- Fourier transforms convert convolution into multiplication; the convention affects factors of 2π, so track the definition used.
Worked example
Evaluate I = ∫ from -∞ to ∞ of dx/(1+x²). Close the contour in the upper half-plane; the integrand has a simple pole at z = i. Its residue is 1/(z+i) evaluated at z=i, which gives 1/(2i). By the residue theorem, I = 2πi × 1/(2i) = π. Jordan's lemma justifies the large semicircle contribution going to zero.
Traps that cost marks
- Choosing a contour that encloses the wrong pole or crosses a branch cut.
- Forgetting the Jacobian in polar or spherical coordinates.
- Assuming every matrix is diagonalisable over the reals.
- Losing the sign of a residue when the contour is clockwise.