Partial Derivative/Examples/2 u + 3 v = sin x, u + 2 v = x cos y
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Example of Partial Derivative
Consider the simultaneous equations:
- $\begin {cases} 2 u + 3 v & = \sin x \\ u + 2 v & = x \cos y \end {cases}$
Then:
- $\map {u_1} {\dfrac \pi 2, \pi} = 3$
Proof
By definition of partial derivative:
- $\map {u_1} {\dfrac \pi 2, \pi} = \valueat {\dfrac {\partial u} {\partial x} } {x \mathop = \frac \pi 2, y \mathop = \pi}$
hence the motivation for the abbreviated notation on the left hand side.
Explicit Method
We have:
- $2 u + 3 v = \sin x$
Thus:
| \(\ds u\) | \(=\) | \(\ds \dfrac {\sin x - 3 v} 2\) | ||||||||||||
| \(\text {(1)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds \dfrac {\partial u} {\partial x}\) | \(=\) | \(\ds \dfrac {\cos x} 2 - \dfrac 3 2 \dfrac {\partial v} {\partial x}\) |
Then we have:
- $u + 2 v = x \cos y$
Thus:
| \(\ds v\) | \(=\) | \(\ds \dfrac {x \cos y - u} 2\) | ||||||||||||
| \(\text {(2)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds \dfrac {\partial v} {\partial x}\) | \(=\) | \(\ds \dfrac {\cos y} 2 - \dfrac 1 2 \dfrac {\partial u} {\partial x}\) |
Substituting $\dfrac {\partial v} {\partial x}$ from $(2)$ into $(1)$ gives:
| \(\ds \dfrac {\partial u} {\partial x}\) | \(=\) | \(\ds \dfrac {\cos x} 2 - \dfrac 3 2 \paren {\dfrac {\cos y} 2 - \dfrac 1 2 \dfrac {\partial u} {\partial x} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {\cos x} 2 - \dfrac {3 \cos y} 4 + \dfrac 3 4 \dfrac {\partial u} {\partial x}\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \dfrac {\partial u} {\partial x} \paren {1 - \dfrac 3 4}\) | \(=\) | \(\ds \dfrac {\cos x} 2 - \dfrac {3 \cos y} 4\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \dfrac {\partial u} {\partial x}\) | \(=\) | \(\ds 2 \cos x - 3 \cos y\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \map {u_1} {\dfrac \pi 2, \pi}\) | \(=\) | \(\ds 2 \map \cos {\dfrac \pi 2} - 3 \cos \pi\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2 \times 0 - 3 \times \paren {-1}\) | Cosine of Right Angle, Cosine of Straight Angle | |||||||||||
| \(\ds \) | \(=\) | \(\ds 3\) | simplifying |
$\blacksquare$
Implicit Method
| \(\ds 2 u + 3 v\) | \(=\) | \(\ds \sin x\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \map {\dfrac \partial {\partial x} } {2 u + 3 v}\) | \(=\) | \(\ds \map {\dfrac \partial {\partial x} } {\sin x}\) | |||||||||||
| \(\text {(1)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds 2 \dfrac {\partial u} {\partial x} + 3 \dfrac {\partial v} {\partial x}\) | \(=\) | \(\ds \cos x\) |
| \(\ds u + 2 v\) | \(=\) | \(\ds x \cos y\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \map {\dfrac \partial {\partial x} } {u + 2 v}\) | \(=\) | \(\ds \map {\dfrac \partial {\partial x} } {x \cos y}\) | |||||||||||
| \(\text {(2)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds \dfrac {\partial u} {\partial x} + 2 \dfrac {\partial v} {\partial x}\) | \(=\) | \(\ds \cos y\) |
Combining $(1)$ and $(2)$ into matrix form:
- $\begin {pmatrix} 2 & 3 \\ 1 & 2 \end {pmatrix} \begin {pmatrix} \dfrac {\partial u} {\partial x} \\ \dfrac {\partial v} {\partial x} \end {pmatrix} = \begin {pmatrix} \cos x \\ \cos y \end {pmatrix}$
Hence by Cramer's Rule:
| \(\ds \dfrac {\partial u} {\partial x}\) | \(=\) | \(\ds \dfrac {\begin {vmatrix} \cos x & 3 \\ \cos y & 2 \end {vmatrix} } {\begin {vmatrix} 2 & 3 \\ 1 & 2 \end {vmatrix} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {2 \cos x - 3 \cos y} {2 \times 2 - 1 \times 3}\) | Definition of Determinant | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2 \cos x - 3 \cos y\) | simplifying |
Hence:
| \(\ds \map {u_1} {\dfrac \pi 2, \pi}\) | \(=\) | \(\ds 2 \map \cos {\dfrac \pi 2} - 3 \cos \pi\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 2 \times 0 - 3 \times \paren {-1}\) | Cosine of Right Angle, Cosine of Straight Angle | |||||||||||
| \(\ds \) | \(=\) | \(\ds 3\) | simplifying |
$\blacksquare$
Sources
- 1961: David V. Widder: Advanced Calculus (2nd ed.) ... (previous) ... (next): $1$ Partial Differentiation: $\S 1$. Introduction: Exercise $14$