Where the Einstein summation notation is used and the product of the vectors, is a dyadic tensor of type ( 2, 0 ), or the Jacobian matrix
42.
If =, the Jacobian matrix is a square matrix, and its determinant, a function of, is the "'Jacobian determinant "'of.
43.
If = 1, is a scalar field and the Jacobian matrix is reduced to a row vector of partial derivatives of i . e . the gradient of.
44.
According to the inverse function theorem, the matrix inverse of the Jacobian matrix of an invertible function is the Jacobian matrix of the " inverse " function.
45.
According to the inverse function theorem, the matrix inverse of the Jacobian matrix of an invertible function is the Jacobian matrix of the " inverse " function.
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However, for a problem size on the order of 2 billion, the Jacobian matrix is likely to contain on the order of a trillion non-zero entries.
47.
Although expressed in terms of coordinates, the result is invariant under Jacobian matrix of an-dimensional vector field in-dimensional space is invariant under any invertible linear transformation.
48.
In vector calculus, the "'Jacobian matrix "'(, ) is the matrix of all first-order partial derivatives of a vector-valued function.
49.
A necessary and sufficient condition for this inverse function to exist is that the determinant of the Jacobian Matrix, often referred to simply as the Jacobian, should be different from zero.
50.
To state the implicit function theorem, we need the Jacobian matrix of " f ", which is the matrix of the partial derivatives of " f ".
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