Deriving Maxwell's Equations

In vector calculus, the gradient of a scalar function $T$, is the vector function $\mathbfit{\nabla}T$, which points in the direction of the maximum increase of $T$, and whose magnitude is the rate of increase of $T$ along this direction. If $T$ is a scalar function of 3 variables $T = T(x, y, z)$, and $\mathbfit{\hat{x},\hat{y},\hat{z}} $ are the unit vectors parallel to $x$, $y$, and $z$ axes, respectively, derive the mathematical definition of $\mathbfit{\nabla}T$ using a theorem on partial derivatives: $dT=\left(\frac{\partial T}{\partial x}\right) dx + \left(\frac{\partial T}{\partial y}\right) dy + \left(\frac{\partial T}{\partial z}\right) dz$.

When the $\mathbfit{\nabla}$ operator ($\mathbfit{\nabla}=\mathbfit{\hat{x}}\frac{\partial}{\partial x}+\mathbfit{\hat{y}}\frac{\partial}{\partial y}+\mathbfit{\hat{z}}\frac{\partial}{\partial z}$) acts on a scalar function, the result is called the gradient. $$\\$$ (a) What is the result called, when $\mathbfit{\nabla}$ acts upon a vector function $\mathbfit F$ via the dot product: $\mathbfit {\nabla} \cdot \mathbfit{F}$? $$\\$$ (b) Using the definition of the $\mathbfit{\nabla}$ operator ($\mathbfit{\nabla}=\mathbfit{\hat{x}}\frac{\partial}{\partial x}+\mathbfit{\hat{y}}\frac{\partial}{\partial y}+\mathbfit{\hat{z}}\frac{\partial}{\partial z}$), derive the mathematical equation for $\mathbfit{\nabla}\cdot \mathbfit {F}$, if $F_x$, $F_y$, and $F_z$ are the projections of $\mathbfit F$ along the $x$, $y$, and $z$ axes, respectively. $$\\$$ (c) Is the result a vector or a scalar?

When the $\mathbfit{\nabla}$ operator ($\mathbfit{\nabla}=\mathbfit{ \hat{x}}{\partial \over{\partial x}}+\mathbfit{ \hat{y}}{\partial \over{\partial y}}+\mathbfit{\hat{z}}{\partial \over{\partial z}}$) acts on a scalar function, the result is called the gradient. $$\\$$ (a) What is the result called when $\mathbfit{\nabla}$ acts upon a vector function $\mathbfit F$ via the cross product: $\mathbfit{\nabla}\times \mathbfit{F}$? $$\\$$ (b) Using the determinant (matrix), derive the mathematical expression for $\mathbfit{\nabla}\times \mathbfit{F}$, if $F_x$, $F_y$, and $F_z$ are the projections of $\mathbfit F$ along the $x$, $y$, and $z$ axes, respectively. $$\\$$ (c) Is the result a vector or a scalar?