List of top Mathematics Questions on Vector Calculus asked in AP ECET BSc Mathematics

If \[ \mathbf{F}=ax\mathbf{i}+by\mathbf{j}+cz\mathbf{k}, \] where $a$, $b$, and $c$ are constants, and $S$ is the surface of the unit sphere, then \[ \iint_{S}\mathbf{F}\cdot\mathbf{n}\,dS=\_ \]

If \[ \overline{a}\ \text{and}\ \overline{b} \] are constant vectors and \[ \overline{r}=x\mathbf{i}+y\mathbf{j}+z\mathbf{k}, \] then \[ \nabla\!\left(\overline{r}\cdot(\overline{a}\times\overline{b})\right)=\_ \]

If \[ \overline{F}=(2x^{2}-3z)\mathbf{i}-2xy\mathbf{j}-4x\mathbf{k}, \] then \[ \int_{V}\nabla\cdot\overline{F}\,dV=\_ \] where $V$ is the closed region bounded by \[ x=0,\quad y=0,\quad z=0,\quad \text{and}\quad x+2y+z=4. \]

If the surface $mx^{2} - 2nyz = (m + 4)x$ is orthogonal to the surface $4x^{2}y + z^{3} = 4$ at the point (1, -1, 2), then the values m and n will be}

Let $u = x + y + z, v = x^{2} + y^{2} + z^{2}, w = yz + zx + xy$. Then the vectors $\nabla u, \nabla v, \nabla w$ are}

Gauss divergence theorem states the relation between

If $r$ is the position vector of any point on a closed surface S that encloses the volume V then $\iint_{S}(\overline{r} \cdot ds)$ is equal to}

If $\overline{r}(t) = ti - t^{2}j + (t - 1)k$ and $\overline{s}(t) = 2t^{2}i + 6tk$ then the value of $\int_{0}^{2}\overline{r} \cdot \overline{s} dt$ is}

If $\overline{r} = xi + yj + zk$ and $r = |\overline{r}| \neq 0$ then div. ($r^{n}\overline{r}$) =}

Let $F = (3xy)i + (2y^{2})j$. The value of the line integral $\int_{C}F \cdot dr = _. $ where C is the arc of the parabola $y = 2x^{2}$ from (0,0) to (1,2).

Let $F = (2xy + z^{3})i + x^{2}j + 3xz^{2}k$. The work done in moving a particle in this field from (1, -2, 1) to (3, 1, 4) is}

If C is the rectangle bounded by the lines $x = 0, x = a, y = 0, y = b$, then the value of $\int_{C}(x^{2} + y^{2})dx - 2xy dy$ is}

The velocity field on an incompressible flow is given by $\overline{v} = (a_1x + a_2y + a_3z)i + (b_1x + b_2y + b_3z)j + (c_1x + c_2y + c_3z)k$ where $a_1 = 2, c_3 = -4$ then the value of $b_2 = _$

If $\overline{a}$ is constant vector then $\nabla \times (\overline{a} \times \overline{r}) = _?$

Find the directional derivative of $f = xyz$ at the point $(1,-1,-2)$ in the direction of the vector $2i-2j+k$ is

The vector field $\vec{V} = e^{x}\sin y~i + e^{x}\cos y~j$ is}

The velocity vector is given as $\overline{V} = 5xyi + 2y^{2}j + 3yz^{2}k$. The divergence of this vector at (1,1,1) is}

If $\phi = 2x^{3}y^{2}z^{4}$ then $\nabla^{2}\phi = $

If $f$ and $F$ are scalar and vector functions respectively, then which of the following is Not Correct?

For a vector $\overline{A}$, which of the following statement is NOT TRUE?

The circulation of $\vec{F}=y\hat{i}+z\hat{j}+x\hat{k}$ around the circle $x^2+y^2=1,\ z=0$ is ____.

In an inner product space $V(F)$, for $a,b\in F$ and $\alpha,\beta\in V$, then $(a\alpha,b\beta)=$

For $\alpha,\beta\in V$, in an inner product space $V(F)$, the triangle inequality states that

If $f(2)=2\hat i-\hat j+2\hat k$ and $f(3)=4\hat i-2\hat j+3\hat k$, then the value of $\int_2^3\left(f\cdot\dfrac{df}{dt}\right)dt$ is

The value of $\iint_S (x^2+2y^2+3z^2)\,ds$ over the sphere $x^2+y^2+z^2=9$ is