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Let $f ∈ C^1(R^n, R)$. If $||Df(x_0)||≠0$ then f increases most in the direction $Df(x_0)$ at $x_0$ and if it is =0, the derivative of f is 0 in any direction.

Could someone try to explain what this means? How can we evaluate the norm of the Jacobian? Is it Euclidean norm? I would really appreciate any help in visualizing what it actually means as well. It's my first time studying this and I would appreciate it if it is explained in the simplest terms possible since I do not have much knowledge on topology and multivariable calculus. Thank you!

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  • $\begingroup$ The norm is unnecessary here anyway, since the condition that the norm is nonzero just equivalent to the condition that the Jacobian is nonzero. Roughly speaking the norm of the Jacobian measures the sensitivity of a function to small perturbations. It's probably the Euclidean norm here but for a general multivariate function it might be the operator norm. $\endgroup$ Commented Jan 10, 2023 at 8:51
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    $\begingroup$ The sign (positive/negative) of the Jacobian tells you if your mapping (locally) is orientation preserving or not. The magnitude (norm) of the Jacobian tells you by what factor your map (locally) blows up an infinitesmial region of space. $\endgroup$ Commented Jan 10, 2023 at 9:12

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The sentence "$f$ increases most in the direction $Df(x_0)$" is misleading and hide numerous technical details. Formally, $Df(x_0)$ is a linear operator, the differential of $f$ at a point, not a vector. The Jacobian matrix of $f \in C^1(R^n, R^m)$ is the linear map representing the differential. If $n=m=1$, the differential is simply what we call the derivative of $f$ and if $m=1$, the Jacobian matrix contains a unique line vector. In the end, in the specific case $f ∈ C^1(R^n, R)$, the direction of maximal increase is given by the transpose of the Jacobian Matrix, $Df(x_0)^T$, or represen. Indeed, the Jacobian matrix is made of partial derivatives, and as $f \in C^1(R^n, R)$, directional derivatives are deduced from the partial derivatives.

So, as explained, $||Df(x_0)||$ should not be understood as a norm of a vector but as a norm of a linear operator, or a matrix in this case. Since it is still a norm in the usual sense, $||Df(x_0)||=0$ implies that this linear operator is the zero operator, i.e. all partial derivatives are $0$, hence all directional derivatives as well. Conversely, if $||Df(x_0)|| \ne 0$, then there is at least one partial derivative that is non-zero. There are many norms on matrices depending on the context, some of them are described here.

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