PYQs Based Test - 10 : Application of Theorems

The value of the surface integral $\int \underset{\mathrm{s}}{\int }\left(9\mathrm{x}\hat{\mathrm{i}}-2\mathrm{y}\hat{\mathrm{j}}-\mathrm{z}\hat{\mathrm{k}}\right)\cdot \hat{n}dS$ over the surface S of the sphere x2 + y2 + z2 = 9, where n is the unit outward normal to the surface element dS, is _______.

Given $\vec{F}=z{\hat{a}}_x+x{\hat{a}}_y+y{\hat{a}}_z$. If S represents the portion of the sphere x² + y² + z² = 1 for z ≥ 0 then ${\int }_s\nabla \times \vec{F}\cdot \overrightarrow{ds}$ is ________

Given a vector u = 1/3≤ft( - y^3̂ i + x^3̂ j + z^3̂ k ) and n̂ as the unit normal vector to the surface of the hemisphere (x² + y² + z² = 1; z ≥ 0), the value of integral $\int \left(\nabla \times u\right)\bullet \hat{n}dS$ evaluated on the curved surface of the hemisphere S is

The value (up to two decimal places) of a line integral ${\int }_{\mathrm{c}}\vec{\mathrm{F}}\left(\vec{\mathrm{r}}\right).\mathrm{d}\vec{\mathrm{r}},\mathrm{for}\mathrm{F}\vec{\mathrm{r}}={\mathrm{x}}^2\hat{\mathrm{i}}+{\mathrm{y}}^2\hat{\mathrm{j}}$ along C which is a straight line joining (0, 0) to (1, 1) is ______

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 system having i, j, and k as unit base vectors.

_s^1/4≤ft( F.n )dA , Where S is the sphere, x² + y² + z² = 1 and n is the outward unit normal vector to the sphere. The value of the surface integral is