Geometric Meaning of the Partial Derivative Could anyone explain the geometric meaning behind the partial derivative?
I know that for a "normal" derivative, its geometric meaning is the slope of the tangent of a curve. From what I understand about the partial derivative, it is the slope of the tangent of a cross section of a function with two or more variables. If anyone could expand upon the definition and clarify, I would be grateful. Thank you.
 A: Think about a surface described by the equation $z = z(x,y)$. At a given point $(x_0,y_0)$, the partial derivative $\partial z/ \partial x$ is the derivative of the function $x \mapsto z(x,y_0)$ with respect to $x$. Of course, the graph of this function $x \mapsto z(x,y_0)$ is a "slice" through the original surface that is cut by the plane $y=y_0$. 
You get similar things in other dimensions. Think about a parametric surface $\mathbf S = \mathbf S(u,v)$, where $\mathbf S$ is a mapping from $\mathbb R^2$ to $\mathbb R^3$. At given parameter values $(u_0,v_0)$, the partial derivative $\partial \mathbf S/ \partial u$ is the derivative of the the curve $u \mapsto \mathbf S(u,v_0)$ with respect to $u$. In other words, it's the derivative of a "constant parameter" curve on the surface.
Generally, a partial derivative is the derivative of a curve that's formed by changing one variable while holding all others constant.
Draw some pictures illustrating the two cases I described, and I think you'll get the idea. Then, once you understand how things work in $\mathbb R^2$ and $\mathbb R^3$, you can start pondering more general situations, if you need to. 
A: For $f:\mathbb{R}^n\to \mathbb{R}$:
$$
\frac{\partial f}{\partial x_i} = \nabla f\cdot e_i
$$
is the directional derivative along the direction of $e_i$, $e_i$ is $1$ in the $i$-th component and $0$ in others. 
If the partial derivative is evaluated at a point $x = (x_1,\ldots,x_n)$, define a mapping: 
$$f_i: \mathbb{R} \to \mathbb{R}, \;t\mapsto f(x_1,\ldots,x_i(t),\ldots,x_n),$$
where the $i$-th component is parametrized by $x_i = a + t$, then 
$$
\frac{\partial f}{\partial x_i} = \frac{d f_i}{d t}
$$
which is the slope of the tangent to the curve $f_i(t)$ at the corresponding point.
