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

Best Answer - Chosen by Voters

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

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.

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.