Gradient

The gradient transforms like a vector under change of basis of the space of variables of

, written as an upside-down triangle and pronounced "del", denotes the vector differential operator.

There can be functions for which partial derivatives exist in every direction but fail to be differentiable.

Furthermore, this definition as the vector of partial derivatives is only valid when the basis of the coordinate system is orthonormal.

[3] In this particular example, under rotation of x-y coordinate system, the above formula for gradient fails to transform like a vector (gradient becomes dependent on choice of basis for coordinate system) and also fails to point towards the 'steepest ascent' in some orientations.

The magnitude of the gradient will determine how fast the temperature rises in that direction.

The steepness of the slope at that point is given by the magnitude of the gradient vector.

The magnitude and direction of the gradient vector are independent of the particular coordinate representation.

[4][5] In the three-dimensional Cartesian coordinate system with a Euclidean metric, the gradient, if it exists, is given by

where ρ is the axial distance, φ is the azimuthal or azimuth angle, z is the axial coordinate, and eρ, eφ and ez are unit vectors pointing along the coordinate directions.

where r is the radial distance, φ is the azimuthal angle and θ is the polar angle, and er, eθ and eφ are again local unit vectors pointing in the coordinate directions (that is, the normalized covariant basis).

We consider general coordinates, which we write as x1, …, xi, …, xn, where n is the number of dimensions of the domain.

If the coordinates are orthogonal we can easily express the gradient (and the differential) in terms of the normalized bases, which we refer to as

of covectors; thus the value of the gradient at a point can be thought of a vector in the original

The best linear approximation to a function can be expressed in terms of the gradient, rather than the derivative.

This equation is equivalent to the first two terms in the multivariable Taylor series expansion of

For the second form of the chain rule, suppose that h : I → R is a real valued function on a subset I of R, and that h is differentiable at the point f(a) ∈ I.

A level surface, or isosurface, is the set of all points where some function has a given value.

If f is differentiable, then the dot product (∇f )x ⋅ v of the gradient at a point x with a vector v gives the directional derivative of f at x in the direction v. It follows that in this case the gradient of f is orthogonal to the level sets of f. For example, a level surface in three-dimensional space is defined by an equation of the form F(x, y, z) = c. The gradient of F is then normal to the surface.

More generally, any embedded hypersurface in a Riemannian manifold can be cut out by an equation of the form F(P) = 0 such that dF is nowhere zero.

Similarly, an affine algebraic hypersurface may be defined by an equation F(x1, ..., xn) = 0, where F is a polynomial.

A (continuous) gradient field is always a conservative vector field: its line integral along any path depends only on the endpoints of the path, and can be evaluated by the gradient theorem (the fundamental theorem of calculus for line integrals).

Conversely, a (continuous) conservative vector field is always the gradient of a function.

, and taking the limit yields a term which is bounded from above by the Cauchy-Schwarz inequality[8]

Suppose f : Rn → Rm is a function such that each of its first-order partial derivatives exist on ℝn.

In curvilinear coordinates, or more generally on a curved manifold, the gradient involves Christoffel symbols:

where gjk are the components of the inverse metric tensor and the ei are the coordinate basis vectors.

Expressed more invariantly, the gradient of a vector field f can be defined by the Levi-Civita connection and metric tensor:[11]

Generalizing the case M = Rn, the gradient of a function is related to its exterior derivative, since

More precisely, the gradient ∇f is the vector field associated to the differential 1-form df using the musical isomorphism