A scalar potential $$\,\,\varphi \,\,$$ has the following gradient: $$\,\,\nabla \varphi = yz\widehat i + xz\widehat j + xy\widehat k.\,\,$$ Consider the integral $$\,\,\int_C {\nabla \varphi .d\overrightarrow r \,\,} $$ on the curve $$\overrightarrow r = x\widehat i + y\widehat j + z\widehat k.\,\,$$ The curve $$C$$ is parameterized as follows: $$\,\,\left\{ {\matrix{ {x = t} \cr {y = {t^2}} \cr {z = 3{t^2}} \cr } \,\,\,\,\,\,\,} \right.$$ and $$1 \le t \le 3.\,\,\,\,\,$$
The value of the integral is _________.

The value of the line integral $$\,\,\oint\limits_C {\overrightarrow F .\overrightarrow r ds,\,\,\,} $$ where $$C$$ is a circle of radius $${4 \over {\sqrt \pi }}\,\,$$ units is ________.

Here, $$\,\,\overrightarrow F x,y = y\widehat i + 2x\widehat j\,\,$$ and $$\,\overrightarrow r $$ is the UNIT tangent vector on the curve $$C$$ at an arc length s from a reference point on the curve. $$\widehat i$$ and $$\widehat j$$ are the basis vectors in the $$X-Y$$ Cartesian reference. In evaluating the line integral, the curve has to be traversed in the counter-clockwise direction.

The value of $$\int\limits_C {\left[ {\left( {3x - 8{y^2}} \right)dx + \left( {4y - 6xy} \right)dy} \right],\,\,} $$ (where $$C$$ is the region bounded by $$x=0,$$ $$y=0$$ and $$x+y=1$$) is ________.

The surface integral $$\,\,\int {\int\limits_s {{1 \over \pi }} } \left( {9xi - 3yj} \right).n\,dS\,\,$$ over the sphere given by $${x^2} + {y^2} + {z^2} = 9\,\,$$ is __________.