Difference between gradient and Jacobian

Could anyone explain in simple words (and maybe with an example) what the difference between the gradient and the Jacobian is?

The gradient is a vector with the partial derivatives, right?

Answer

These are two particular forms of matrix representation of the derivative of a differentiable function f, used in two cases:

  • when f:RnR, then for x in Rn, gradx(f):=[fx1fx2fxn]|x is the matrix 1×n of the linear map Df(x) exprimed from the canonical base of Rn to the canonical base of R (=(1)…). Because in this case this matrix would have only one row, you can think about it as the vector
    f(x):=(fx1,fx2,,fxn)|xRn.
    This vector f(x) is the unique vector of Rn such that Df(x)(y)=f(x),y for all yRn (see Riesz representation theorem), where , is the usual scalar product
    (x1,,xn),(y1,,yn)=x1y1++xnyn.
  • when f:RnRm, then for x in Rn, Jacx(f)=[f1x1f1x2f1xnf2x1f2x2f2xnfmx1fmx2fmxn]|x is the matrix m×n of the linear map Df(x) exprimed from the canonical base of Rn to the canonical base of Rm.

For example, with f:R2R such as f(x,y)=x2+y you get grad(x,y)(f)=[2x1] (or f(x,y)=(2x,1)) and for f:R2R2 such as f(x,y)=(x2+y,y3) you get Jac(x,y)(f)=[2x103y2].

Attribution
Source : Link , Question Author : Math_reald , Answer Author : Paul Wintz

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