Vector calculus was developed from the theory of quaternions by J. Willard Gibbs and Oliver Heaviside near the end of the 19th century, and most of the notation and terminology was established by Gibbs and Edwin Bidwell Wilson in their 1901 book, Vector Analysis.
Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from point to point.
The three basic vector operators are:[3] Also commonly used are the two Laplace operators: A quantity called the Jacobian matrix is useful for studying functions when both the domain and range of the function are multivariable, such as a change of variables during integration.
Given a differentiable function f(x, y) with real values, one can approximate f(x, y) for (x, y) close to (a, b) by the formula The right-hand side is the equation of the plane tangent to the graph of z = f(x, y) at (a, b).
For a continuously differentiable function of several real variables, a point P (that is, a set of values for the input variables, which is viewed as a point in Rn) is critical if all of the partial derivatives of the function are zero at P, or, equivalently, if its gradient is zero.
The different cases may be distinguished by considering the eigenvalues of the Hessian matrix of second derivatives.
By Fermat's theorem, all local maxima and minima of a differentiable function occur at critical points.
These structures give rise to a volume form, and also the cross product, which is used pervasively in vector calculus.
Vector calculus can be defined on other 3-dimensional real vector spaces if they have an inner product (or more generally a symmetric nondegenerate form) and an orientation; this is less data than an isomorphism to Euclidean space, as it does not require a set of coordinates (a frame of reference), which reflects the fact that vector calculus is invariant under rotations (the special orthogonal group SO(3)).
This structure simply means that the tangent space at each point has an inner product (more generally, a symmetric nondegenerate form) and an orientation, or more globally that there is a symmetric nondegenerate metric tensor and an orientation, and works because vector calculus is defined in terms of tangent vectors at each point.
Grad and div generalize immediately to other dimensions, as do the gradient theorem, divergence theorem, and Laplacian (yielding harmonic analysis), while curl and cross product do not generalize as directly.
Geometric algebra is mostly used in generalizations of physics and other applied fields to higher dimensions.
From this point of view, grad, curl, and div correspond to the exterior derivative of 0-forms, 1-forms, and 2-forms, respectively, and the key theorems of vector calculus are all special cases of the general form of Stokes' theorem.
From the point of view of both of these generalizations, vector calculus implicitly identifies mathematically distinct objects, which makes the presentation simpler but the underlying mathematical structure and generalizations less clear.