Chain Rule for Partial Derivatives
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Theorem
Let $F: \R^2 \to \R$ be a real-valued function of $2$ variables.
Let $X: \R^2 \to \R$ and $Y: \R^2 \to \R$ also be real-valued functions of $2$ variables.
Let $F = \map f {x, y}$ be such that:
\(\ds x\) | \(=\) | \(\ds \map X {u, v}\) | ||||||||||||
\(\ds y\) | \(=\) | \(\ds \map Y {u, v}\) |
Then:
- $F = \map F {u, v}$
and:
\(\ds \dfrac {\partial F} {\partial u}\) | \(=\) | \(\ds \dfrac {\partial f} {\partial x} \dfrac {\partial X} {\partial u} + \dfrac {\partial f} {\partial y} \dfrac {\partial Y} {\partial u}\) | ||||||||||||
\(\ds \dfrac {\partial F} {\partial v}\) | \(=\) | \(\ds \dfrac {\partial f} {\partial x} \dfrac {\partial X} {\partial v} + \dfrac {\partial f} {\partial y} \dfrac {\partial Y} {\partial v}\) |
This article is complete as far as it goes, but it could do with expansion. In particular: Conditions on continuity and/or differentiability need to be incorporated here You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Expand}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Corollary 1
Let $F = \map f {x, y}$ be a real-valued function from $\R^2$ to $\R$.
Let $x = \map X t$ and $y = \map Y t$ be real functions.
Then:
- $F = \map F t$
and:
- $\dfrac {\d F} {\d t} = \dfrac {\partial F} {\partial x} \dfrac {\d x} {\d t} + \dfrac {\partial F} {\partial y} \dfrac {\d Y} {\d t}$
Corollary 2
Let $F = \map f {x, y}$ be a real-valued function from $\R^2$ to $\R$.
Let $y = \map Y x$ be a real function.
Then:
- $\dfrac {\d F} {\d x} = \dfrac {\partial F} {\partial x} + \dfrac {\partial F} {\partial y} \dfrac {\d Y} {\d x}$
Proof
This theorem requires a proof. In particular: Discussed here: https://www.ma.imperial.ac.uk/~jdg/AECHAIN.PDF which is appropriate considering this result I'm reading from a reference book written by staff at Imperial. Use Cauchy-Riemann Equations (a special case) as a basis for this, and make that page dependent upon this. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Sources
- 1969: J.C. Anderson, D.M. Hum, B.G. Neal and J.H. Whitelaw: Data and Formulae for Engineering Students (2nd ed.) ... (previous) ... (next): $4.$ Mathematics: $4.4$ Differential calculus: $\text {(v)}$ Partial differentiation: $\text {(c)}$