The vector that is NOT perpendicular to the vectors $$\,\,\left( {i + j + k} \right)\,\,$$ and $$\,\left( {i + 2j + 3k} \right)\,\,$$ is _________.

The magnitude of the directional derivative of the function $$f\left( {x,y} \right) = {x^2} + 3{y^2}$$ in a direction normal to the circle $$\,{x^2} + {y^2} = 2,$$ at the point $$(1,1),$$ is

$$A$$ vector is defined as $$f = y\widehat i + x\widehat j + z\widehat k\,\,$$. Where $$\widehat i,\widehat j,$$ and $$\widehat k$$ are unit vectors in cartesian $$(x, y, z)$$ coordinate system. The surface integral - -over the closed surface $$S$$ of a cube with vertices having the following coordinates: $$(0,0,0), (1, 0, 0), (0, 1, 0), (0,0,1), (1, 0, 1), (1,1,1), (0, 1, 1), (1, 1, 0)$$ is _______.

For a vector $$E,$$ which one of the following statements is NOT TRUE?

A sphere of unit radius is centered at the origin. The unit normal at a point $$(x, y, z)$$ on the surface of the sphere is the vector.

If a vector $$\overrightarrow R \left( t \right)$$ has a constant magnitude then

The direction of vector $$A$$ is radially outward from the origin, with $$\left| A \right| = K\,{r^n}$$ where $${r^2} = {x^2} + {y^2} + {z^2}$$ and $$K$$ is constant. The value of $$n$$ for which $$\nabla .A = 0\,\,$$ is

A scalar field is given by $$f = {x^{2/3}} + {y^{2/3}},$$ where $$x$$ and $$y$$ are the Cartesian coordinates. The derivative of $$'f'$$ along the line $$y=x$$ directed away from the origin at the point $$(8, 8)$$ is