Q:

Why is a gradient field a special case of a vector field?

My calculus manual suggests a gradient field is just a special case of a vector field. That implies that there are vector fields that there are not gradient fields. The gradient field is composted of a vector and each $\mathbf{i}$, $\mathbf{j}$, $\mathbf{k}$ component (using 3 dimensions) is multiplied by a scalar that is a partial derivative. Is this because the scalar may be of the form that there is no antiderivative of? Perhaps of a function that can not be differentiated? Maybe a vector field that has sharp turns but I can't come up with any that I can't find a derivative for. Does anybody have any so I can get some intuition of the problem?

A:

If you have a gradient field $(X=\partial_xf,Y=\partial_yf,Z=\partial_zf)$,
by using the relation ${\partial^2f\over{\partial x\partial y}}={{\partial^2f}\over{\partial y\partial x}}$ you obtain $\partial_yX=\partial_xY$. This is not true for every vector field.
For example $(x,x^2y^2,z)$ is not a gradient vector field.

A:

A vector field is just a function $\mathbf{F}:\mathbb{R}^n\to\mathbb{R}^n$. (Or the domain can be some subset $D$ of $\mathbb{R}^n$, not all of it, of course. To avoid making this remark repetitively, I'll just stick with $\mathbb{R}^n$, as an example). From an intuitive/geometrical point of view, it is best visualized as having a vector attached to (or starting from) each point of the domain $D$. Given a multivariable function $f:\mathbb{R}^n\to\mathbb{R}$, its gradient $\nabla f$ is precisely that kind of a function, as it acts from $\mathbb{R}^n$ to $\mathbb{R}^n$, so it is a vector field.
What the book says is that not all vector fields can be obtained in this way. Intuitively, it should make sense: the fact that the components of $\nabla f$ are derived from the same original function $f$ should result in some relationships among them. If you put random functions as components, most likely they wouldn't be related in that special way. It's like siblings: being the descendants of the same parents they have some things in common, something that random people wouldn't.
In more mathematical terms, one of such special relationships is the condition of equality of mixed partial derivatives. Say, for a function of two variables $f:\mathbb{R}^2\to\mathbb{R}$, we must have $\displaystyle \frac{\partial^2f}{\partial y\partial x}=\frac{\partial^2f}{\partial x\partial y}$, assuming these derivatives are continuous. Therefore, if $\displaystyle \mathbf{F}=\nabla f=\left\langle\frac{\partial f}{\partial x},\frac{\partial f}{\partial y}\right\rangle=\langle M,N\rangle$ is its gradient, then we have to have
$$\color{blue}{\frac{\partial M}{\partial y}}=\frac{\partial^2f}{\partial y\partial x}=\frac{\partial^2f}{\partial x\partial y}=\color{blue}{\frac{\partial N}{\partial x}}.$$
So being the gradient is very special! This condition allows you to construct vector fields that are not gradients. Pretty much anything you would put into a vector field at random almost surely wouldn't be a gradient. For example, $\mathbf{F}(x,y)=\langle 2x+3y,4x+5y\rangle$ isn't a gradient, because
$$\frac{\partial M}{\partial y}=3\neq4=\frac{\partial N}{\partial x}.$$

