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The polynomial map and its Jacobian
Example
Let . Then is totally differentiable everywhere and
Facts & Assumptions
Given: The polynomial map .
The function is differentiable everywhere with derivative for positive natural (For a natural the function is differentiable everywhere with derivative ; for it is the constant , with derivative ; for a natural the function is differentiable at every with derivative ; consequently every polynomial function is differentiable at every real, with the derivative computed term by term).
Continuous partial derivatives imply total differentiability with Jacobian derivative (If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative).
Verification
The four partial derivatives are , , , and , by [L1] and derivative algebra.
These polynomial partial derivatives are continuous everywhere, so [L2] gives total differentiability and identifies the derivative with the displayed Jacobian.
Directly, the increment remainder is , whose norm divided by tends to zero, agreeing with step 2.1.
Depends on
- If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative
- The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case
- For a natural $n \ge 1$ the function $x \mapsto x^{n}$ is differentiable everywhere with derivative $\iota(n)\,x^{\,n-1}$; for $n = 0$ it is the constant $1$, with derivative $0$; for a natural $n \ge 1$ the function $x \mapsto x^{-n}$ is differentiable at every $x \ne 0$ with derivative $-\iota(n)\,x^{-n-1}$; consequently every polynomial function is differentiable at every real, with the derivative computed term by term
- Sums, scalar multiples, products and quotients: $(f+g)'(c) = f'(c) + g'(c)$, $(\alpha f)'(c) = \alpha f'(c)$, $(fg)'(c) = f'(c)g(c) + f(c)g'(c)$, and $(f/g)'(c) = \bigl(f'(c)g(c) - f(c)g'(c)\bigr)/g(c)^{2}$ when $g(c) \ne 0$
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
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Sources
- J. Lebl, Basic Analysis I, §8.4 (standard reference, not scraped)