List of top Mathematical Physics Questions

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?

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?

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?

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?