STAT312 Lecture : Multidimensional calculus.pdf
5 Apr 2014
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The jacobian matrix is the matrix jf (x) = This arrangement of partial derivatives ensures that the chain rule is easily z 1 then x (30. 1) f y 1 if x 1 (writing y x for jf (x) etc. ) The jacobian matrix of vector whose transpose is the gradient: : r: this matrix h (x) has ( In most cases these di erentiations can be carried out in either order, so that the hessian is a symmetric matrix. R then the directional derivative at a in the direction v (with kvk = 1) is. = 0f (a)v (a + v) (a) lim. Proof: using (30. 1), the limit is (a + v) | =0 (a + v) (a + v) (a + v) We say a is a stationary point of. R if f (a) = 0; this means the directional derivative. : r is 0 in all directions v. : r region under the graph of the function x.
