Engineering Mat...

If C is the path along the curve y = x² – 4x + 4 from (0, 4) to (2, 0), then \(\mathop \oint \nolimits_C \left( {y\hat i - 3x\hat j} \right) \cdot \overrightarrow {dr} \) is

The value of the integral

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

The value of \(\mathop \smallint \limits_C^\; \left( {2x + 3y} \right)dx - \left( {3x - 4y} \right)dy\) where c is the circle with radius as 1 and centre at origin.

 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.

If S is the surface of the sphere x² + y² + z² = a², then the value of

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

For vectors \(\vec F = 3xy\hat i - {y^2}\hat j\) and \(\vec R = x\hat i + y\hat j\), the value of \(\mathop \smallint \limits_C \vec F.d\vec R\) on the curve C (y = 2x²) in the x-y plane from (0, 0) to (1, 2) is

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 ________.