What is a Gradient (Maths)?

A gradient is a measure of how steep a slope is. It quantifies the change in elevation between two points on a surface, and is represented by a line on a graph. The steeper the gradient, the greater the change in elevation.

The word “gradient” comes from the Latin gradus, meaning “step”. In mathematics, a gradient is typically denoted by the symbol ∇ (nabla). The nabla symbol originates from Greek, where it represents a triangle. This symbol is often seen in physics and engineering applications, as it represents the direction of maximum change (in this case, of elevation).

Gradients can be positive or negative; they can also be zero. A positive gradient indicates that the slope is increasing (getting steeper), while a negative gradient indicates that the slope is decreasing (getting less steep). A zero gradient means that there is no change in elevation over time – the surface is level.

Gradients are used extensively in mathematics and science. In calculus, for example, gradients are used to find maxima and minima of functions; they are also used in vector fields to determine directions of flow. In physics, gradients are important in many applications involving waves and wave-like phenomena (such as heat diffusion).

To calculate a gradient, we first need to define a function. For our purposes, let’s say that the function is f(x) = x2. We can then use the following formula:

∇f(x) = [f'(x), f”(x)]

where f'(x) denotes the derivative of f with respect to x, and f”(x) denotes the second derivative of f with respect to x. In this case, we get:

∇f(x) = [2x, 0]

This tells us that the gradient of our function at any point x is equal to 2x (i.e., it is positive and increasing as x increases).

We can also use the gradient to find the equation of a tangent line to our function at any point x. To do this, we simply need to plug the value of x into our formula for the gradient:

∇f(x) = [2x, 0]

This tells us that the equation of the tangent line at any point x is y = 2x.