Vector Calculus

Vector field $\vec{V}$ is defined as $\vec{V} = 3x^2yz \hat{i} - 5xy \hat{j} + 6yz^2 \hat{k}$ The curl of $\vec{V}$ at point (2,-1,1) is

Consider a velocity vector, $\vec{V}$ in (x, y, z) coordinates given below. Pick one or more CORRECT statements(s) from the choices given below. $\vec{V} = u\hat{x} + v\hat{y}$

Consider a velocity vector, $\vec{V}$ in ( $\mathrm{x}, \mathrm{y}, \mathrm{z}$ ) coordinates given below. Pick one or more CORRECT statement(s) from the choices given below: $$ \v...

A vector field $\vec{p}$ and a scalar field $r$ are given by $\vec{p} = (2x^2 – 3xy + z^2 ) \hat{i} + ( 2y^2 – 3yz + x^2 ) \hat{j} + (2z^2 – 3xz + x^2 ) \hat{k}$ $r = 6x^2 + 4y^2 –...

A vector field $\vec{p}$ and a scalar field $r$ are given by: $\vec{p} = (2x^2 - 3xy + z^2) \hat{i} + (2y^2 - 3yz + x^2) \hat{j} + (2z^2 - 3xz + x^2) \hat{k}$ $r = 6x^2 + 4y^2 - z^...

Let $\phi$ be a scalar field, and $\mathbf{u}$ be a vector field. Which of the following identities is true for $\text{div}(\phi\mathbf{u})$?

Let 𝜙 be a scalar field, and 𝒖 be a vector field. Which of the following identities is true for div(𝜙𝒖)?

A function is defined in Cartesian coordinate system as $f(x, y) = xe^y$. The value of the directional derivative of the function (in integer) at the point (2, 0) along the directi...

If C represents a line segment between (0,0,0) and (1,1,1) in Cartesian coordinate system, the value (expressed as integer) of the line integral $\int_C [(y+z)dx+(x+z)dy+ (x + y)dz...

What is curl of the vector field 2x²yi + 5z²j – 4yzk ?

The velocity field in a flow system is given by v = 2i + (x + y)j + (xyz)k. The acceleration of the fluid at (1, 1, 2) is

The value (up to two decimal places) of a line integral $\int_{C} \mathbf{F}(\mathbf{r}) \cdot d\mathbf{r}$, for $\mathbf{F}(\mathbf{r})=x^{2}\hat{i}+y^{2}\hat{j}$ along C which is...

The divergence of the vector field $$\,V = {x^2}i + 2{y^3}j + {z^4}k\,\,$$ at $$x=1, y=2, z=3$$ is ________.

The directional derivative of the field $$u(x, y, z)=$$ $${x^2} - 3yz$$ in the direction of the vector $$\left( {\widehat i + \widehat j - 2\widehat k} \right)\,\,$$ at point $$(2,...

For a scalar function $$f(x,y,z)=$$ $${x^2} + 3{y^2} + 2{z^2},\,\,$$ the gradient at the point $$P(1,2,-1)$$ is

For a scalar function $$\,f\left( {x,y,z} \right) = {x^2} + 3{y^2} + 2{z^2},\,\,$$ the directional derivative at the point $$P(1,2,-1)$$ in the direction of a vector $$\widehat i -...

The velocity vector is given as $${\mkern 1mu} \vec V = 5xy\widehat i + 2{y^2}\widehat j + 3y{z^2}\widehat k.{\mkern 1mu} {\mkern 1mu} $$ The divergence of this velocity vector at...

The directional derivative of $$\,\,f\left( {x,y,z} \right) = 2{x^2} + 3{y^2} + {z^2}\,\,$$ at the point $$P(2,1,3)$$ in the direction of the vector $${\mkern 1mu} \vec a = \wideha...

The line integral $$\int {\,\,V.dr\,\,} $$ of the vector function $$V\left( r \right) = 2xyz\widehat i + {x^2}z\widehat j + {x^2}y\widehat k\,\,$$ from the origin to the point $$P(...

Value of the integral $$\,\,\oint {xydy - {y^2}dx,\,\,} $$ where, $$c$$ is the square cut from the first quadrant by the line $$x=1$$ and $$y=1$$ will be (Use Green's theorem to ch...

The vector field $$\,F = x\widehat i - y\widehat j\,\,$$ (where $$\widehat i$$ and $$\widehat j$$ are unit vectors) is

The directional derivative of the following function at $$(1, 2)$$ in the direction of $$(4i+3j)$$ is : $$f\left( {x,y} \right) = {x^2} + {y^2}$$

For the function $$\phi = a{x^2}y - {y^3}$$ to represent the velocity potential of an ideal fluid, $${\nabla ^2}\,\,\phi $$ should be equal to zero. In that case, the value of $$'a...

The directional derivative of the function $$f(x, y, z) = x + y$$ at the point $$P(1,1,0)$$ along the direction $$\overrightarrow i + \overrightarrow j $$ is

The derivative of $$f(x, y)$$ at point $$(1, 2)$$ in the direction of vector $$\overrightarrow i + \overrightarrow j $$ is $$2\sqrt 2 $$ and in the direction of the vector $$ - 2\o...