Let \(d\) be the spatial dimension. \(\alpha = (\alpha_1, \cdots, \alpha_d) \in \mathbb{N}_0^d\).

\[\lvert \alpha \rvert = \sum_{i = 1}^d \alpha_i\]

The factorization of a multi-index is equal to the product of the factorization of each component index:

\[\alpha ! = \prod_{i = 1}^d \alpha_i !\]

The multi-index is distributed to each coordinate component of \(x \in \mathbb{R}^d\):

\[x^{\alpha} = \prod_{i = 1}^d x_i^{\alpha_i}\]

The \(\lvert \alpha \rvert\)-fold mixed derivative with respect to \(x\):

\[\begin{array}{rl} \partial_x^{\alpha} & = \prod_{i = 1}^d \partial_{x_i}^{\alpha_i} \\ \partial_x^{\alpha} x^{\beta} & = \frac{\beta !}{(\beta - \alpha) !} x^{\beta - \alpha} \quad (\alpha \leq \beta) \end{array}\]

where \(\beta \in \mathbb{N}_0^d\), \(\alpha \leq \beta\) means \(\alpha_i \leq \beta_i\) for \(i = 1, \cdots, d\) and \(\beta - \alpha\) is the normal subtraction of two vectors. When \(\beta \equiv \alpha\),

\[\partial_x^{\alpha} x^{\alpha} = \alpha !\]

The combination for multi-index:

\[C_{\alpha}^{\beta} = \prod_{i = 1}^d C_{\alpha_i}^{\beta_i} \quad (\beta \leq \alpha)\]

Proposition 1. (Leibniz formula for \(\lvert \alpha \rvert\)-fold partial derivatives)

\[\partial_x^{\alpha} (uv) = \sum_{\beta \leq \alpha} C_{\alpha}^{\beta} (\partial_x^{\beta} u) (\partial_x^{\alpha - \beta} v),\]

where \(\alpha\) and \(\beta\) are multi-indices and \(x \in \mathbb{R}^d\).

Proof 1. When \(d=1\), it is the classical Leibiniz formula in calculus.

2. Assume it holds for \(d=n\). Then for \(d=n+1\), we have

\[\begin{equation*} \pdiff_x^{\alpha}(uv) =\left( \prod_{i=1}^{n+1}\pdiff_{x_i}^{\alpha_i} \right)(uv) = \pdiff_{x_1}^{\alpha_1} \left( \prod_{i=2}^{n+1}\pdiff_{x_i}^{\alpha_i} \right) \end{equation*}\]

Let \(\tilde{\alpha}=(\alpha_2,\cdots,\alpha_{n+1})\) and \(\tilde{x}=(x_2,\cdots,x_{n+1})\),

\[\begin{equation*} \begin{split} \text{Above} &= \pdiff_{x_1}^{\alpha_1} \pdiff_{\tilde{x}}^{\tilde{\alpha}}(uv) = \pdiff_{x_1}^{\alpha_1}\left( \sum_{\tilde{\beta}\leq\tilde{\alpha}}C_{\tilde{\alpha}}^{\tilde{\beta}}(\pdiff_x^{\tilde{\beta}}u)(\pdiff_x^{\tilde{\alpha}-\tilde{\beta}}v) \right) \\ &= \sum_{\tilde{\beta}\leq\tilde{\alpha}}C_{\tilde{\alpha}}^{\tilde{\beta}} \underbrace{\pdiff_{x_1}^{\alpha_1}\left[(\pdiff_x^{\tilde{\beta}}u)(\pdiff_x^{\tilde{\alpha}-\tilde{\beta}}v)\right]}_{\text{Apply the Leibniz formula for $d=1$}} \\ &= \sum_{\tilde{\beta}\leq\tilde{\alpha}}C_{\tilde{\alpha}}^{\tilde{\beta}}\left[ \sum_{\beta_1\leq\alpha_1}C_{\alpha_1}^{\beta_1}(\pdiff_{x_1}^{\beta_1}\pdiff_{\tilde{x}}^{\tilde{\beta}}u)(\pdiff_{x_1}^{\alpha_1-\beta_1}\pdiff_{\tilde{x}}^{\tilde{\alpha}-\tilde{\beta}}v) \right] \\ &= \sum_{\tilde{\beta}\leq\tilde{\alpha}}\sum_{\beta_1\leq\alpha_1}C_{\tilde{\alpha}}^{\tilde{\beta}}C_{\alpha_1}^{\beta_1}(\pdiff_x^{\beta}u)(\pdiff_x^{\alpha-\beta}v) \\ &= \sum_{\beta\leq\alpha}C_{\alpha}^{\beta}(\pdiff_x^{\beta}u)(\pdiff_x^{\alpha-\beta}v). \end{split} \end{equation*}\]

3. Apply the principle of mathematical induction, the proposition is proved.

Proposition 2. (Taylor expansion for multi-dimensional functions) Let \(f : X \rightarrow \mathbb{R}\) be a function from \(C^m (X)\) with \(X \subset \mathbb{R}^d\). Let \(x_0 \in X\) be an expansion center. The Taylor expansion around \(x_0\) is

\[f (x) = \sum_{\text{$\begin{array}{c} \alpha \in \mathbb{N}_0^d\\ \lvert \alpha \rvert \leq m \end{array}$}} (x - x_0)^{\alpha} \frac{1}{\alpha !} \partial_x^{\alpha} f (x_0) + R_r,\]

where \(R_r\) is the high order remainder.

Proposition 3. The total number of terms in the \(m\)-th order Taylor expansion for a function defined on \(X \subset \mathbb{R}^d\) is \(C_{m + d}^d\).

Proof The number of terms in the Taylor expansion for \(\abs{\alpha}=k\) is equivalent to the number of ways for distributing the \(k\) times of partial derivatives to the \(d\) coordinate components, some of which may be assigned no derivatives at all. Hence the answer is \(C_{k+d-1}^{d-1}\). The total number of terms in the Taylor expansion is

\[\begin{equation*} \begin{split} \sum_{k=0}^{m}C_{k+d-1}^{d-1} &= C_{d-1}^{d-1} + C_d^{d-1} + \cdots + C_{m+d-1}^{d-1} \\ \text{Replace $C_{d-1}^{d-1}$ with $C_d^d$,} \\ &= C_d^d + C_d^{d-1} + C_{d+1}^{d-1} + \cdots + C_{m+d-1}^{d-1} \\ \text{Apply $C_n^{m-1} + C_n^m = C_{n+1}^m$,} \\ &= C_{d+1}^d + C_{d+1}^{d-1} + \cdots + C_{m+d-1}^{d-1} \\ &= C_{m+d}^d. \end{split} \end{equation*}\]

Comment For Sobolev norms in 1, there are also partial derivatives (in the weak sense) of different orders:

\[\lVert u \rVert_{m, p} = \left( \sum_{\lvert \alpha \rvert \leq m} \lVert \partial^{\alpha} u \rVert_p^p \right)^{1 / p} \quad (1 \leq p < \infty) .\]

References

  1. Robert Alexander Adams. Sobolev Spaces (Pure and applied mathematics, a series of monographs and textbooks).  PURE AND APPLIED MATHEMATICS: A series of Monographs and Textbooks ISBN: 9780120441501. Academic Press, June 1975. ↩