Search results
Results from the WOW.Com Content Network
The above formulas for the derivative of a vector function rely on the assumption that the basis vectors e 1, e 2, e 3 are constant, that is, fixed in the reference frame in which the derivative of a is being taken, and therefore the e 1, e 2, e 3 each has a derivative of identically zero.
In vector calculus, the derivative of a vector function y with respect to a vector x whose components represent a space is known as the pushforward (or differential), or the Jacobian matrix. The pushforward along a vector function f with respect to vector v in R n is given by d f ( v ) = ∂ f ∂ v d v . {\displaystyle d\mathbf {f} (\mathbf {v ...
Specifically, the divergence of a vector is a scalar. The divergence of a higher-order tensor field may be found by decomposing the tensor field into a sum of outer products and using the identity, where is the directional derivative in the direction of multiplied by its magnitude. Specifically, for the outer product of two vectors,
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 ...
Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional Euclidean space, . [1] The term vector calculus is sometimes used as a synonym for the broader subject of multivariable calculus, which spans vector calculus as well as partial differentiation and multiple integration.
The directional derivative of a scalar function = (,, …,) along a vector = (, …,) is the function defined by the limit [4] = (+) (). This definition is valid in a broad range of contexts, for example where the norm of a vector (and hence a unit vector) is undefined.
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity of change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point.
The directional derivative of a scalar function = (,, …,) along a vector = (, …,) is the function defined by the limit [1] = (+) (). This definition is valid in a broad range of contexts, for example where the norm of a vector (and hence a unit vector) is undefined.