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t. e. In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions f and g in terms of the derivatives of f and g. More precisely, if is the function such that for every x, then the chain rule is, in Lagrange's notation, or, equivalently, The chain rule may also be expressed in ...
The triple product rule, known variously as the cyclic chain rule, cyclic relation, cyclical rule or Euler's chain rule, is a formula which relates partial derivatives of three interdependent variables. The rule finds application in thermodynamics, where frequently three variables can be related by a function of the form f (x, y, z) = 0, so ...
by the change of variables: in these steps: Replace by and apply the chain rule to get. Replace and by and to get. Replace and by and and divide both sides by to get. Replace by and divide through by to yield the heat equation. Advice on the application of change of variable to PDEs is given by mathematician J. Michael Steele: [1]
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 ...
If the direction of derivative is not repeated, it is called a mixed partial derivative. If all mixed second order partial derivatives are continuous at a point (or on a set), f is termed a C 2 function at that point (or on that set); in this case, the partial derivatives can be exchanged by Clairaut's theorem:
Miscellanea. v. t. e. In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices. It collects the various partial derivatives of a single function with respect to many variables, and/or of a multivariate function with respect to a single variable, into vectors and matrices that ...
The total derivative is a linear combination of linear functionals and hence is itself a linear functional. The evaluation measures how much points in the direction determined by at , and this direction is the gradient. This point of view makes the total derivative an instance of the exterior derivative. Suppose now that is a vector-valued ...
Jacobi's formula. In matrix calculus, Jacobi's formula expresses the derivative of the determinant of a matrix A in terms of the adjugate of A and the derivative of A. [1] If A is a differentiable map from the real numbers to n × n matrices, then. where tr (X) is the trace of the matrix X and is its adjugate matrix.