Engineering Mat...

Divergence value of a function $x^{3}y\overrightarrow{i}-\left(z^2-2y\right)\overrightarrow{j}+5y^{2}z\overrightarrow{k}$ at x = 2, y = 3 and z = 4 is

If $\overline{r}=x\hat{i}+y\hat{j}+z\hat{k}$ and r = |r̅|, then ∇² [log r] = ?

Consider an incompressible flow velocity given as

$\overrightarrow{V}=\left(2x+3y+4z\right)\hat{i}+\left(5x+cy+6z\right)\hat{j}+\left(8x+9y\right)\hat{k}$ 

Let $\overrightarrow{a}=\lambda \hat{i}-9\hat{j}-\hat{k},\overrightarrow{b}=3\hat{i}+3\hat{j}+\hat{k}and\overrightarrow{c}=4\hat{i}+2\hat{j}+\hat{k}$. The value of λ for which the vector $\overrightarrow{a}$ is perpendicular to $\overrightarrow{b}\times \overrightarrow{c}$ is ________.

The value of $\underset{C}{\int }\left[\left(2x^2-y\right)dx+3x{y}^{2}dy\right]$ and C is the curve x² = 4y and y² = 4x is _____.

The circulation of A̅ = yî + zĵ + xk̂ around the circle x² + y² = 1, z = 0 is

If f = x²yz and g = xy – 3z², then the value ∇ ⋅ (∇f × ∇g) of (1, -1, 2) is ______ 

The vector $\overrightarrow{V}=\left(x+y+az\right)i+\left(bx+2y-z\right)j++\left(-x+cy+2z\right)k$ is irrotational. Where a, b and c are constants. Find the divergence of the vector $\overrightarrow{V}$.

The directional derivative of ϕ = x²yz + 4 xz² at (1, -2, -1) in the direction 2î - ĵ - 2k [upto 2 decimals]

If F = 3y î – xz ​ĵ + yz² k̂ and S is the surface of the paraboloid 2z = x² + y² bounded by z = 8, evaluate $\int \underset{s}{\overset{}{\int }}\left(\mathrm{\nabla }\times F\right).ds$