Recent questions tagged vector-identities

The directional derivative of $f(x,y,z) = xyz$ at point $(-1,1,3)$ in the direction of vector $\hat{i} – 2 \hat{j} +2 \hat{k}$ is$3\hat{i} – 3 \hat{j} - \hat{k} \\$$- \df...

A vector field is defined as $$\overrightarrow{f}\left ( x,y,z \right )=\dfrac{x}{\left [ x^{2}+y^{2}+z^{2} \right ]^{\frac{3}{2}}}\widehat{i}\:+\:\dfrac{y}{\left [ x^{2}...

Given a vector $\overrightarrow{u} = \dfrac{1}{3} \big(-y^3 \hat{i} + x^3 \hat{j} + z^3 \hat{k} \big)$ and $\hat{n}$ as the unit normal vector to the surface of the hemi...

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

For a position vector $\overrightarrow{r} = x \hat{i}+y \hat{j} + z\hat{k}$ the norm of the vector can be defined as $\mid \overrightarrow{r} \mid = \sqrt{x^2+y^2+z^2}$. ...

The value of the integralover the closed surface $S$ bounding a volume $V$, where $\overrightarrow{r} = x \hat{i} + y \hat{j}+z \hat{k}$ is the position vector and $\over...

The divergence of the vector $-yi+xj$ is ________.

For the vector $\vec{V}=2yz\hat{i}+3xz \hat{j}+4xy \hat{k}$, the value of $\bigtriangledown$. $(\bigtriangledown \times \vec{V})$ is ________.

A point P $(1, 3,−5)$ is translated by $2\hat{i}+3\hat{j}-4\hat{k}$ and then rotated counter clockwise by $90^\circ $ about the $z$-axis. The new position of the point i...

The value of the line integral $\oint_{c}^{ }\overline{F}.{\overline{r}}'ds$ ,where $C$ is a circle of radius $\dfrac{4}{\sqrt{\pi }}$ units is ________Here, $\overline{F...

A scalar potential $\varphi$ has the following gradient: $\bigtriangledown \varphi =yz\hat{i}+xz\hat{j}+xy\hat{k}$ . Consider the integral $\int_{c}^{ }\bigtriangledown \...

The figure below represents a triangle $PQR$ with initial coordinates of the vertices as $P(1,3)$, $Q(4,5)$ and $R(5,3.5)$. The triangle is rotated in the $X$-$Y$ plane a...

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 o...

Consider a velocity field $\overrightarrow{V}=k(y\hat{i}+x\hat{k})$ , where $K$ is a constant. The vorticity, $Ω_Z$ , is$-K$$K$$-K/2$$K/2$

Divergence of the vector field $x_2z\hat{i}+xy\hat{j}-yz\hat{k}$ at $(1,-1,1)$ is$0$$3$$5$$6$

Curl of vector $\overrightarrow{F}=x^2z^2\hat{i}-2xy^2z\hat{j}+2y^2z^3\hat{k}$ is$(4yz^3+2xy^2)\hat{i}+2x^2z\hat{j}-2y^2z\hat{k}$$(4yz^3+2xy^2)\hat{i}-2x^2z\hat{j}-2y^2z\...

Which one of the following describes the relationship among the three vectors, $\hat{i}+\hat{j}+\hat{k}$ , $2\hat{i}+3\hat{j}+\hat{k}$ and $5\hat{i}+6\hat{j}+4\hat{k}$ ?T...