Derivative of Scalar Triple Product of Vector-Valued Functions
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Theorem
Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be differentiable vector-valued functions in Cartesian $3$-space.
The derivative of their scalar triple product is given by:
- $\map {\dfrac \d {\d x} } {\mathbf a \cdot \paren {\mathbf b \times \mathbf c} } = \dfrac {\d \mathbf a} {\d x} \cdot \paren {\mathbf b \times \mathbf c} + \mathbf a \cdot \paren {\dfrac {\d \mathbf b} {\d x} \times \mathbf c} + \mathbf a \cdot \paren {\mathbf b \times \dfrac {\d \mathbf c} {\d x} }$
Proof
\(\ds \map {\dfrac \d {\d x} } {\mathbf a \cdot \paren {\mathbf b \times \mathbf c} }\) | \(=\) | \(\ds \dfrac {\d \mathbf a} {\d x} \cdot \paren {\mathbf b \times \mathbf c} + \mathbf a \cdot \map {\dfrac \d {\d x} } {\mathbf b \times \mathbf c}\) | Derivative of Dot Product of Vector-Valued Functions | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\d \mathbf a} {\d x} \cdot \paren {\mathbf b \times \mathbf c} + \mathbf a \cdot \paren {\dfrac {\d \mathbf b} {\d x} \times \mathbf c + \mathbf b \times \dfrac {\d \mathbf c} {\d x} }\) | Derivative of Vector Cross Product of Vector-Valued Functions | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\d \mathbf a} {\d x} \cdot \paren {\mathbf b \times \mathbf c} + \mathbf a \cdot \paren {\dfrac {\d \mathbf b} {\d x} \times \mathbf c} + \mathbf a \cdot \paren {\mathbf b \times \dfrac {\d \mathbf c} {\d x} }\) | Dot Product Distributes over Addition |
$\blacksquare$
Also see
- Derivative of Dot Product of Vector-Valued Functions
- Derivative of Vector Cross Product of Vector-Valued Functions
- Derivative of Product of Real Function and Vector-Valued Function
Sources
- 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text {III}$: The Differentiation of Vectors: $2$. Differentiation of Sums and Products
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 22$: Formulas involving Derivatives: $22.25$