Jacobian

1. Jacobian wiki

In vector calculus, the Jacobian matrix (/dʒəˈkoʊbiən/,^[1][2][3] /dʒɪ-, jɪ-/) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables as input as the number of vector components of its output, its determinant is referred to as the Jacobian determinant. Both the matrix and (if applicable) the determinant are often referred to simply as the Jacobian in literature.^[4]

2.  Vector-Valued function wiki

In vector calculus, the Jacobian matrix (/dʒəˈkoʊbiən/,^[1][2][3] /dʒɪ-, jɪ-/) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables as input as the number of vector components of its output, its determinant is referred to as the Jacobian determinant. Both the matrix and (if applicable) the determinant is often referred to simply as the Jacobian in literature.^[4]

3. In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space.^[1] For instance, a vector field in the plane can be visualised as a collection of arrows with a given magnitude and direction, each attached to a point in the plane. 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 one point to another point.