Vector Calculus · Engineering Mathematics · GATE ECE

A surface is given by $z^2=2 x^2-y^2$ and $\vec{n}$ and $-\vec{n}$ are unit normal vectors to the surface at the point $\vec{P}=\hat{i}+\sqrt{2} \hat{k}$.

Which of the following vectors can be $\vec{n}$, where $\hat{i}, \hat{j}$ and $\hat{k}$ and are the unit vectors along $x, y$ and $z$ axes, respectively?

Let $\rho(x, y, z, t)$ and $u(x, y, z, t)$ represent density and velocity, respectively, at a point $(x, y, z)$ and time $t$. Assume $\frac{\partial \rho }{\partial t}$ is continuous. Let $V$ be an arbitrary volume in space enclosed by the closed surface $S$ and $\hat{n}$ be the outward unit normal of $S$. Which of the following equations is/are equivalent to $\frac{\partial \rho }{\partial t} + \nabla \cdot(\rho u) = 0$?

Let $${v_1} = \left[ {\matrix{ 1 \cr 2 \cr 0 \cr } } \right]$$ and $${v_2} = \left[ {\matrix{ 2 \cr 1 \cr 3 \cr } } \right]$$ be two vectors. The value of the coefficient $$\alpha$$ in the expression $${v_1} = \alpha {v_2} + e$$, which minimizes the length of the error vector e, is

The rate of increase, of a scalar field $$f(x,y,z) = xyz$$, in the direction $$v = (2,1,2)$$ at a point (0,2,1) is

For a vector field $\vec{A}$, which one of the following is FALSE?

Let $F_1$, $F_2$, and $F_3$ be functions of $(x, y, z)$. Suppose that for every given pair of points A and B in space, the line integral $\int\limits_C (F_1 dx + F_2 dy + F_3 dz)$ evaluates to the same value along any path C that starts at A and ends at B. Then which of the following is/are true?

For the solid $S$ shown below, the value of $\iiint_S x d x d y d z$ (rounded off to two decimal places) is $\_\_\_\_$ .

Note: $$C$$ and $${S_C}$$ refer to any closed contour and any surface whose boundary is $$C.$$