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

Let C be the curve, which is the union of two line segments, the first going from (0, 0) to (3, 4) and the second segment going from (3, 4) to (6, 0). Compute the integral \(\mathop \smallint \nolimits_C \left( {3dy - 4dx} \right)\)

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 the region below z = 4 – xy and above the region in the xy-plane defined by 0 ≤ x ≤ 2, 0 ≤ y ≤ 1 is _______

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

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