Engineering Mat...

If \({\rm{\vec F}} = {\rm{x\;\vec i}} + {\rm{y\;\vec j}} + {\rm{z\;\vec k}}\) and s is the closed surface of x² + y² + z² = a². Then \(\mathop \int\!\!\!\int \nolimits_{\rm{s}}^{} {\rm{\vec F}} \cdot {\rm{\hat n\;ds}}\) is

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

Evaluate the integral \(I = \mathop \oint \limits_c \left( {{e^x}dx + 2ydy - dz} \right)\) over the curve C: x² + y² = 4, z = 2

The value of the line integral \(\frac{2}{\pi }\mathop \oint \limits_\gamma \left( { - {y^3}dx + {x^3}dy} \right)\), where γ is the circle x² + y² = 1 oriented counter clockwise, is ________.

Evaluate the line integral of \(\vec F = \left( {x + y} \right)\hat i + \left( {2x - z} \right)\hat j + \left( {y + 2} \right)\hat k\) along the boundary of triangle whose vertices are (2, 0, 0), (0, 3, 0) and (0, 0, 6)

Evaluate the line integral of vector field F = sin y î + x (1 + cos x) ĵ along the circular path given by x² + y² = a², z = 0

Find the value of work done on a semi sphere \({x^2} + {y^2} + {z^2} = 4\) bounded by a curve C, given that \(F = {x^2}i + \left( {{y^2} + x} \right)j + \left( {{z^2}} \right)k\)