List of top Engineering Mathematics Questions on Calculus asked in GATE XE

Consider the function \[ f(x, y) = x^2y + 2xy^2 - 2x^2y^2. \] Then which one of the following statements is correct?

Let \( C \) be the positively oriented boundary of the domain bounded by the curves \( y = 2x^2 \) and \( y^2 = 4x \). Then the value of the line integral \[ \oint_C \left( 2y^2 + 2xy + 4y \right) dx + \left( x^2 + 4xy + 8x \right) dy \] is equal to:

Consider the second order Partial Differential Equation (PDE)
\[ 4x^2 \frac{\partial^2 u}{\partial x^2} + 4(x + y) \frac{\partial^2 u}{\partial x \partial y} + (x^2 + y^2) \frac{\partial^2 u}{\partial y^2} - u = 0. \] Then which one of the following statements is correct?