List of top Mathematical Physics Questions asked in GATE PH

If \( F_1(Q, q) = Qq \) is the generating function of a canonical transformation from \((p, q)\) to \((P, Q)\), then which one of the following relations is correct?

Given that \( A^\alpha \) and \( B_\beta \) (\(\alpha, \beta = 1, 2, 3, \ldots, n\)) are contravariant and covariant vectors, respectively, and by convention, any repeated indices are summed over. Which of the following expressions is/are tensors?

The Fourier transform and its inverse transform are respectively defined as:\[ \tilde{f}(\omega) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} f(x) e^{i \omega x} dx \]and\[ f(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} \tilde{f}(\omega) e^{-i \omega x} d\omega \]Consider two functions \( f \) and \( g \). Another function \( f * g \) is defined as:\[ (f * g)(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} f(y) g(x - y) dy \]Which of the following relation is/are true?Note: Tilde (~) denotes the Fourier transform.

Consider a vector field \( \vec{F} = (2xz + 3y^2)\hat{y} + 4yz^2\hat{z} \). The closed path (\( \Gamma \): \( A \rightarrow B \rightarrow C \rightarrow D \rightarrow A \)) in the \( z = 0 \) plane is shown in the figure.

\( \oint_\Gamma \vec{F} \cdot d\vec{l} \) denotes the line integral of \( \vec{F} \) along the closed path \( \Gamma \). Which of the following options is/are true?

Consider two matrices: \( P = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix} \) and \( Q = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \). Which of the following statements is/are true?

The complex function \( e^{-\left(\frac{2}{z-1}\right)} \) has:

Consider a volume integral
\(I=\int_V\nabla^2(\frac{1}{r})dv\)
over a volume \( V \), where \( r = \sqrt{x^2 + y^2 + z^2} \). Which of the following statements is/are correct?

A contour integral is defined as \[ I_n = \oint_C \frac{dz}{(z - n)^2 + \pi^2} \] where \( n \) is a positive integer and \( C \) is the closed contour, as shown in the figure, consisting of the line from \( -100 \) to \( 100 \) and the semicircle traversed in the counter-clockwise sense. The value of \( \sum_{n=1}^5 I_n \) (in integer) is \(\underline{\hspace{2cm}}\).

If \( y_n(x) \) is a solution of the differential equation \[ y'' - 2xy' + 2ny = 0 \] where \( n \) is an integer and the prime (\( ' \)) denotes differentiation with respect to \( x \), then acceptable plot(s) of \( \psi_n(x) = e^{-x^2/2} y_n(x) \), is(are)

A function \( f(t) \) is defined only for \( t \geq 0 \). The Laplace transform of \( f(t) \) is \[ \mathcal{L}(f; s) = \int_0^\infty e^{-st} f(t) \, dt \] whereas the Fourier transform of \( f(t) \) is \[ \tilde{f}(\omega) = \int_0^\infty f(t) e^{-i\omega t} \, dt. \] The correct statement(s) is(are)

P and Q are two Hermitian matrices and there exists a matrix \( R \), which diagonalizes both of them, such that \( RPR^{-1} = S_1 \) and \( RQR^{-1} = S_2 \), where \( S_1 \) and \( S_2 \) are diagonal matrices. The correct statement(s) is(are)