vector-calculus

Let φ be a scalar function. Then, ∇φ is

The divergence of the curl of a vector field is

The directional derivative of the function $f$ given below at the point $(1,0)$ in the direction of $\frac{1}{2}(\hat{i} + \sqrt{3} \hat{j})$ is ________ (rounded off to 1 decimal...

Consider a velocity field $\vec{V} = 3z\hat{i} + 0\hat{j} + Cx \hat{k}$, where C is a constant. If the flow is irrotational, the value of C is _________ (rounded off to 1 decimal p...

The directional derivative of the function $f$ given below at the point $(1,0)$ in the direction of $\frac{1}{2}(\hat{i}+\sqrt{3} \hat{j})$ is _______ (Rounded off to 1 decimal pla...

The divergence of the curl of a vector field is

The value of the surface integral ∬_S z dx dy where S is the external surface of the sphere x² + y² + z² = R² is

A vector field B(x, y, z) = x î + y ĵ – 2z k is defined over a conical region having height h = 2, base radius r = 3 and axis along z, as shown in the figure. The base of the cone...

Given a function $\rm ϕ = \frac{1}{2} (x^2 + y^2 + z^2) $ in three-dimensional Cartesian space, the value of the surface integral ∯ S n̂ . ∇ϕ dS where S is the surface of a sphere...

Consider a cube of unit edge length and sides parallel to co-ordinate axes, with its centroid at the point (1, 2, 3). The surface integral $\int_A \vec{F}.d\vec{A}$ of a vector fie...

Given a vector $\vec{u} = \frac{1}{3}(-y^3 \hat{i} + x^3 \hat{j} + z^3 \hat{k})$ and $\hat{n}$ as the unit normal vector to the surface of the hemisphere $(x^2 + y^2 + z^2 = 1; z \...

The value of the integral $\oiint_S \vec{r} \cdot \vec{n} \, dS$ over the closed surface S bounding a volume V, where $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ is the position vec...

For a position vector $\vec{r} = x\hat{i}+y\hat{j}+z\hat{k}$ the norm of the vector can be defined as $|\vec{r}| = \sqrt{x^2 + y^2 + z^2}$. Given a function $\phi = \ln|\vec{r}|$,...

For the vector $$\overrightarrow V = 2yz\widehat i + 3xz\widehat j + 4xy\widehat k,$$ the value of $$\,\nabla .\left( {\nabla \times \overrightarrow \nabla } \right)\,\,$$ is _____...

The surface integral $$\int {\int\limits_s {F.ndS} } $$ over the surface $$S$$ of the sphere $${x^2} + {y^2} + {z^2} = 9,$$ where $$\,F = \left( {x + y} \right){\rm I} + \left( {x...

A scalar potential $$\,\,\varphi \,\,$$ has the following gradient: $$\,\,\nabla \varphi = yz\widehat i + xz\widehat j + xy\widehat k.\,\,$$ Consider the integral $$\,\,\int_C {\na...

The value of the line integral $$\,\,\oint\limits_C {\overrightarrow F .\overrightarrow r ds,\,\,\,} $$ where $$C$$ is a circle of radius $${4 \over {\sqrt \pi }}\,\,$$ units is __...

The surface integral $$\,\,\int {\int\limits_s {{1 \over \pi }} } \left( {9xi - 3yj} \right).n\,dS\,\,$$ over the sphere given by $${x^2} + {y^2} + {z^2} = 9\,\,$$ is __________.

The value of $$\int\limits_C {\left[ {\left( {3x - 8{y^2}} \right)dx + \left( {4y - 6xy} \right)dy} \right],\,\,} $$ (where $$C$$ is the region bounded by $$x=0,$$ $$y=0$$ and $$x+...

Let $$\phi $$ be an arbitrary smooth real valued scalar function and $$\overrightarrow V $$ be an arbitrary smooth vector valued function in a three dimensional space. Which one of...

Curl of vector $$\,V\left( {x,y,x} \right) = 2{x^2}i + 3{z^2}j + {y^3}k\,\,$$ at $$x=y=z=1$$ is

Curl of vector $$\,\,\overrightarrow F = {x^2}{z^2}\widehat i - 2x{y^2}z\widehat j + 2{y^2}{z^3}\widehat k\,\,$$ is

Divergence of the vector field $${x^2}z\widehat i + xy\widehat j - y{z^2}\widehat k\,\,$$ at $$(1, -1, 1)$$ is

The integral $$\,\,\oint\limits_C {\left( {ydx - xdy} \right)\,\,} $$ is evaluated along the circle $${x^2} + {y^2} = {1 \over 4}\,$$ traversed in counter clockwise direction. The...

The following surface integral is to be evaluated over a sphere for the given steady velocity vector field $$F = xi + yj + zk$$ defined with respect to a Cartesian coordinate syste...

For the spherical surface $${x^2} + {y^2} + {z^2} = 1,$$ the unit outward normal vector at the point $$\left( {{1 \over {\sqrt 2 }},{1 \over {\sqrt 2 }},0} \right)$$ is given by

The divergence of the vector field $$\,3xz\widehat i + 2xy\widehat j - y{z^2}\widehat k$$ at a point $$(1,1,1)$$ is equal to

The directional derivative of the scalar function $$f(x, y, z)$$$$ = {x^2} + 2{y^2} + z\,\,$$ at the point $$P = \left( {1,1,2} \right)$$ in the direction of the vector $$\,\overri...

The divergence of the vector field $$\left( {x - y} \right)\widehat i + \left( {y - x} \right)\widehat j + \left( {x + y + z} \right)\widehat k$$ is

The expression curl $$\left( {grad\,f} \right)$$ where $$f$$ is a scalar function is

If $$\overrightarrow V $$ is a differentiable vector function and $$f$$ is sufficienty differentiable scalar function then curl $$\left( {f\overrightarrow V } \right) = $$ _______.