Engineering Mat...

Which of the following represent the Green's theorem 

The value of

\(\mathop \smallint \nolimits_S^{} \left( {x + z} \right)dydz + \left( {y + z} \right)dxdz + \left( {x + y} \right)dxdy,\) 

Where ‘S’ is the surface of the sphere

x² + y² + z² = 4, is ______

If S be any closed surface, evaluate \(\mathop \smallint \limits_S^\; Curl\;\vec F.\vec {ds}\)

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

The value of the integral

\(\mathop \oint \nolimits_s \vec r.\vec n\;ds\)

Suppose C is any curve from (0, 0, 0) to (1, 1, 1) and \(\vec F\left( {x,\;y,\;z} \right) = \left( {4z + 5y} \right)\hat i + \left( {3z + 5x} \right)\hat j + \left( {3y + 4x} \right)\hat k\). Compute the line integral \(\mathop \smallint \nolimits_C \vec F \cdot \overrightarrow {dr} \)

 Evaluate \(\mathop \smallint \nolimits_C \vec F \cdot \overrightarrow {dr} \) where \(\vec F\left( {x,\;y,\;z} \right) = x\hat i + y\hat j + 3\left( {{x^2} + {y^2}} \right)\hat k\) and C is the boundary of the part of the paraboid where z² = 64 – x² – y² which lies above the xy-plane and C is oriented counter clockwise when viewed from above.

The volume of an object expressed as spherical co-ordinate is given by

V = \(\mathop \smallint \limits_0^{2 \pi } \mathop \smallint \limits_0^{\frac{{2 \pi }}{3}} \mathop \smallint \limits_0^2 {{\rm{r}}^3}{\rm{co}}{{\rm{s}}^2}\phi {\rm{drd}}\phi {\rm{d\theta }}\)