In mathematics, the gradient of a scalar-valued function of multiple variables is a vector field. At each point in the function's domain, the gradient vector points in the direction of greatest rate of increase of the function, and its magnitude represents the slope of the graph along this direction.

Formal Definition

Consider a scalar-valued differentiable function of multiple variables:

f : ℝⁿ → ℝ

Where ℝⁿ denotes an n-dimensional Euclidean space. The gradient of f is denoted as \(\nabla_f (x)\) (also sometimes written as grad f ). It is defined as the unique vector field whose dot product with an arbitrary unit vector v at each point x equals the directional derivative of f along v:

\(\nabla_f (x) \cdot v = D_v f(x)\)

Components and Notation

If f is given in standard Cartesian coordinates (\(x₁, x₂, ... x_n\)), then the gradient is represented by the following vector:

\(\nabla_f = \left( \frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2}, \dots, \frac{\partial f}{\partial x_n} \right)\)

Where \(\frac{\partial f}{\partial x_i}\) denotes the partial derivative of f with respect to the i-th variable.


Gradients play a fundamental role in various fields of mathematics and applied sciences:

  • Optimization: They are used in gradient descent algorithms to find minima of functions.
  • Physics: In electromagnetism, the gradient of the electric potential represents the negative of the electric field. The gradient of pressure is important in fluid mechanics.
  • Image processing: Gradients are used in edge detection algorithms, where significant changes in pixel brightness indicate edges.
  • Machine Learning: Gradient information is vital for training algorithms in various machine learning techniques.

Geometric Interpretation

The gradient can be visualized as follows:

  • Direction: The gradient vector points in the direction of the steepest ascent of the function.
  • Magnitude: The magnitude of the gradient vector represents the rate of change in the function in that direction.

Gradient of Vector Fields

While the above primarily describes the gradient of a scalar field, the concept can be extended to vector fields.

  • Jacobian Matrix: The gradient of a vector field is represented by its Jacobian matrix, containing all first-order partial derivatives.