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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.



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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).

Currently studying multivariable calculus
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It's good to cite your sources. – Rahul Sep 5 '12 at 7:07

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 $y=f(a)-\frac{1}{f'(a)}(x-a)$.


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.

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