# Gradient Function

A Gradient Function is a differentiable scalar field function that returns a Vector that points in the direction of the greatest rate of increase in the scalar field.

**AKA:**Grad, Gradient Operation.**Context:**- It can be created by a Gradient Creation System.
- It can range from being an Exact Gradient Function to being an Approximate Gradient Function.

**See:**Gradient Estimation, Gradient Descent Algorithm, Conjugate Gradient Method, Differentiable Function, Scalar-Valued Function, Partial Derivative, Vector-Valued Function, Slope, Tangent, Real-Valued Function, Vector (Mathematics), Partial Derivative, Vector-Valued Function.

## References

### 2016

- (Wikipedia, 2016) ⇒ https://en.wikipedia.org/wiki/gradient Retrieved:2016-10-19.
- In mathematics, the
**gradient**is a generalization of the usual concept of derivative to functions of several variables. If*f*(*x*_{1}, ...,*x*_{n}) is a differentiable, real-valued function of several variables, its gradient is the vector whose components are the*n*partial derivatives of*f*. It is thus a vector-valued function.Similarly to the usual derivative, the gradient represents the slope of the tangent of the graph of the function. More precisely, the gradient points in the direction of the greatest rate of increase of the function, and its magnitude is the slope of the graph in that direction. The components of the gradient in coordinates are the coefficients of the variables in the equation of the tangent space to the graph. This characterizing property of the gradient allows it to be defined independently of a choice of coordinate system, as a vector field whose components in a coordinate system will transform when going from one coordinate system to another.

The Jacobian is the generalization of the gradient for vector-valued functions of several variables and differentiable maps between Euclidean spaces or, more generally, manifolds. A further generalization for a function between Banach spaces is the Fréchet derivative.

- In mathematics, the

### 2009

- http://www.ucl.ac.uk/Mathematics/geomath/level2/fvec/fv7.html
- Suppose we have a scalar function that depends on three space coordinates, x, y and z. Let's call it T. For example it could be the temperature in the room you're in now.
Since T depends on those three variables we can ask the question: how does T change when we change one or more of those variables?

And as always, the answer is found by differentiating the function. In this case, because the function depends on more than one variable, we're talking partial differentiation. (Have a look at the MathHelp notebook on Partial Differentiation if you're not sure about this.)

Now if we differentiate T with respect to x, that tells us the change of T in the x-direction. That is therefore the i-component of the gradient of T.

- Suppose we have a scalar function that depends on three space coordinates, x, y and z. Let's call it T. For example it could be the temperature in the room you're in now.