Difference between Slope and Gradient It has been a few years since studying contour maps. 
Often I hear slope and gradient interchangeably in describing steepness.
Does anyone know any good definitions and analogies of slope and gradient.
Thanks,
Amanda
 A: Contour maps graph level curves of a given function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$. A parametrization $(x(t),y(t))$ of the level curve $f(x,y)=k$ satisfies $f(x(t),y(t))=k$. Differentiating with respect to $t$ yields the following by the chain rule,
$$ \frac{\partial f}{\partial x}\frac{dx}{dt}+ \frac{\partial f}{\partial y}\frac{dy}{dt}=0 $$
Therefore, the gradient $\nabla f = \langle \frac{\partial f}{\partial x},\frac{\partial f}{\partial y} \rangle$ is perpendicular to the tangent vector $\langle x',y'\rangle$ of the curve when we compare them at a particular point.
Comparing slope to $\nabla f$ directly is geometrically questionable since $y=f(x)$ is a graph whereas the natural context for the gradient is in the study of contours. For a graph $z = f(x,y)$ the tangent plane has normal $\pm \langle \partial_x f,\partial_y f,-1 \rangle$. The natural analogue to the normal of the tangent plane would be the slope or perhaps the direction vector of the normal line $y=f(a)-\frac{1}{f'(a)}(x-a)$. 
Slogans:
1.) the gradient points in the direction for you to level-up
2.) the derivative is the slope of the tangent line
To really understand you must separate the concepts of the graph of a function and level curves. These are related but they are not the same object.
A: 
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A gradient is a vector, and slope is a scalar. Gradients really become meaningful in multivarible functions, where the gradient is a vector of partial derivatives. With single variable functions, the gradient is a one dimensional vector with the slope as its single coordinate (so, not very different to the slope at all).

Source(s):
Currently studying multivariable calculus

A: 
Often I hear slope and gradient interchangeably in describing steepness.

This is because gradient and slope can mean the same thing. This depends on which part of the world you live in.

Gradient: (Mathematics) The degree of steepness of a graph at any point.
Slope:  The gradient of a graph at any point.
Source: Oxford Dictionaries

Gradient also has another meaning:

Gradient: (Mathematics) The vector formed by the operator ∇ acting on a scalar function at a given point in a scalar field.
Source: Oxford Dictionaries

Be aware that gradient can mean two different things in mathematics.
A: As gradient is a vector and the gradient of a scaler is a vector ie it drives a scaler in a direction; slope is a scaler ( scaler can be positive and negative) yet it drives a line ( y= mx+ c ) with unique steepness in upward or downward in a direction ( deviation depends upon change in perpendicular per unit base ie tan(theta) which is proportional to theta and up and down  direction by + and - signs) unlike a vector which has finite length (magnitude)
