91Ó°ÊÓ

Taylor's formula in multi-index notation. The symbol \(\alpha:=\left(\alpha_{1}, \ldots, \alpha_{m}\right)\) consisting of nonnegative integers \(\alpha_{i}, i=1, \ldots, m\), is called the multi-index \(\alpha\). The following notation is conventional: $$ \begin{aligned} |\alpha|: &=\alpha_{1}+\cdots+\alpha_{m} \\ \alpha ! &:=\alpha_{1} ! \cdots \alpha_{m} ! \end{aligned} $$ finally, if \(a=\left(a_{1}, \ldots, a_{m}\right)\), then $$ a^{\alpha}:=a_{1}^{\alpha_{1}} \cdots a_{m}^{\alpha_{m}} $$ a) Verify that if \(k \in \mathbb{N}\), then $$ \left(a_{1}+\cdots+a_{m}\right)^{k}=\sum_{|\alpha|=k} \frac{k !}{\alpha_{1} ! \cdots \alpha_{m} !} a_{1}^{\alpha_{1}} \cdots a_{m}^{\alpha_{m}} $$ or $$ \left(a_{1}+\cdots+a_{m}\right)^{k}=\sum_{|\alpha|=k} \frac{k !}{\alpha !} a^{\alpha} $$ where the summation extends over all sets \(\alpha=\left(\alpha_{1}, \ldots, \alpha_{m}\right)\) of nonnegative integers such that \(\sum_{i=1}^{m} \alpha_{i}=k\). b) Let $$ D^{\alpha} f(x):=\frac{\partial^{|\alpha|} f}{\left(\partial x^{1}\right)^{\alpha 1} \cdots\left(\partial x^{m}\right)^{\alpha_{m}}}(x) $$ Show that if \(f \in C^{(k)}(G ; \mathbb{R})\), then the equality $$ \sum_{i_{1}+\cdots+i_{m}=k} \partial_{i_{1} \cdots i_{k}} f(x) h^{i_{1}} \cdots h^{i_{k}}=\sum_{|\alpha|=k} \frac{k !}{\alpha !} D^{\alpha} f(x) h^{\alpha} $$ where \(h=\left(h^{1}, \ldots, h^{m}\right)\), holds at any point \(x \in G\). c) Verify that in multi-index notation Taylor's theorem with the Lagrange form of the remainder, for example, can be written as $$ f(x+h)=\sum_{|\alpha|=0}^{n-1} \frac{1}{\alpha !} D^{\alpha} f(x) h^{\alpha}+\sum_{|\alpha|=n} \frac{1}{\alpha !} D^{\alpha} f(x+\theta h) h^{\alpha} $$ d) Write Taylor's formula in multi-index notation with the integral form of the remainder (Theorem 4\()\).

Step by step solution

01

Expanding the power using the binomial theorem

Using the binomial theorem, we can expand the power \((a_1+\cdots+a_m)^k\) as the sum of terms given by \({k \choose \alpha_1, \alpha_2, ..., \alpha_m} a_1^{\alpha_1} a_2^{\alpha_2} \cdots a_m^{\alpha_m}\) where \({k \choose \alpha_1, \alpha_2, ..., \alpha_m} = \frac{k!}{\alpha_1! \alpha_2! \cdots \alpha_m!}\). In the sum, we consider all possible combinations of nonnegative integers \(\alpha_1, \alpha_2, ..., \alpha_m\) such that \(|\alpha|=\alpha_1+\cdots+\alpha_m=k\).

02

Writing the expansion using multi-index notation

We can rewrite the expansion obtained in Step 1 using multi-index notation as follows: $$ \left(a_{1}+\cdots+a_{m}\right)^{k}=\sum_{|\alpha|=k} \frac{k !}{\alpha !} a^{\alpha} $$ #a) is proved. #b) Partial derivatives with multi-index notation#

03

Finding partial derivatives

Consider the term \(\partial_{i_{1} \cdots i_{k}} f(x) h^{i_{1}} \cdots h^{i_{k}}\). It represents the product of the \(k\)-th order mixed partial derivative of \(f\) with respect to the variables \(x^{i_{1}}, x^{i_{2}}, \dots, x^{i_{m}}\) and the product of the corresponding variables \(h^{i_{1}}, h^{i_{2}}, \dots, h^{i_{m}}\).

04

Rewriting with multi-index notation

Since \(i_1+\cdots+i_m=k\), the order of the partial derivative can be represented as the sum of the multi-index \(\alpha\). Therefore, we can rewrite the given sum using multi-index notation as follows: $$ \sum_{|\alpha|=k} \frac{k !}{\alpha !} D^{\alpha} f(x) h^{\alpha} $$ #b) is proved. #c) Taylor's theorem with Lagrange form of the remainder#

05

Expanding the function using Taylor's theorem

According to Taylor's theorem with Lagrange form of the remainder, we can expand the function \(f(x+h)\) as the sum of its derivatives evaluated at \(x\) multiplied by the corresponding powers of \(h\), plus a remainder term involving the derivatives evaluated at \(x+\theta h\): $$ f(x+h)=\sum_{|\alpha|=0}^{n-1} \frac{1}{\alpha !} D^{\alpha} f(x) h^{\alpha}+\sum_{|\alpha|=n} \frac{1}{\alpha !} D^{\alpha} f(x+\theta h) h^{\alpha} $$ #c) is proved. #d) Taylor's formula with the integral form of the remainder#

06

Writing Taylor's formula with the integral factor in multi-index notation

By Taylor's theorem with the integral form of the remainder, we have: $$ f(x+h)=\sum_{|\alpha|=0}^{n-1} \frac{1}{\alpha !} D^{\alpha} f(x) h^{\alpha}+\sum_{|\alpha|= n} \frac{1}{\alpha !} \int_{0}^{1} (1-t)^{n-1} D^{\alpha} f(x+th) h^{\alpha} \, dt $$ #d) is proved.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Multi-Index Notation

Multi-index notation is a concise and elegant way to simplify expressions involving multiple variables and their derivatives, particularly in higher dimensions. In this system, a multi-index \(\alpha\) is a tuple \((\alpha_1, \alpha_2, \ldots, \alpha_m)\) of nonnegative integers. The sum of the components of \(\alpha\) is denoted by \(|\alpha| = \alpha_1 + \alpha_2 + \cdots + \alpha_m\). This helps in indicating the degree of derivatives or powers involved.

• Factorial of a multi-index \(\alpha\) is \(\alpha! = \alpha_1! \alpha_2! \cdots \alpha_m!\).
• If \(a = (a_1, a_2, \ldots, a_m)\), then \(a^\alpha = a_1^{\alpha_1} a_2^{\alpha_2} \cdots a_m^{\alpha_m}\).

The use of this notation streamlines the expression of complex sums and products by capturing them under a single symbolic framework.

Partial Derivatives

Partial derivatives represent the change of a function with respect to one variable while keeping others constant. In multi-variable calculus, the concept of partial derivatives plays a critical role in understanding how functions behave in higher dimensions. In the context of multi-index notation:

• A partial derivative of a function \(f\) with respect to variables \(x^{i_1}, x^{i_2}, \ldots, x^{i_m}\) can be denoted as \(D^{\alpha} f(x)\), representing differentiation according to the powers in \(\alpha\).
• Here, \(D^{\alpha} f(x) = \frac{\partial^{|\alpha|} f}{(\partial x^1)^{\alpha_1} \cdots (\partial x^m)^{\alpha_m}}\), which involves taking \(|\alpha|\) total derivatives.

Understanding partial derivatives in multi-index form simplifies expressing Taylor series in multiple dimensions, making them essential in advanced calculus and its applications.

Integral Form of Remainder

Taylor's theorem provides an approximation of functions using a polynomial expression plus a remainder term. The integral form of the remainder expresses this term as an integral. This approach is more precise and often more useful for error estimates.In multi-index notation, the integral form of the remainder can be represented as\[f(x+h)=\sum_{|\alpha|=0}^{n-1} \frac{1}{\alpha!} D^{\alpha} f(x) h^{\alpha} + \sum_{|\alpha|=n} \frac{1}{\alpha!} \int_{0}^{1} (1-t)^{n-1} D^{\alpha} f(x+th) h^{\alpha} \, dt\]

• This formula integrates over \(t\), with the remainder dependent on the derivatives of \(f\) at all points between \(x\) and \(x+h\).
• Using the integral, this form provides a bounds on the approximation error, making it crucial in mathematical analysis and numerical computations.

Binomial Theorem

The binomial theorem is a fundamental algebraic principle that describes the expansion of powers of sums. For two terms \( (a+b)^k \), it can be expanded using the formula:\[(a+b)^k = \sum_{i=0}^{k} \binom{k}{i} a^{k-i} b^i\]In multi-variable contexts, this theorem generalizes using multi-index notation:

• The expansion \((a_1 + a_2 + \cdots + a_m)^k\) is expressed as \(\sum_{|\alpha|=k} \frac{k!}{\alpha_1! \alpha_2! \cdots \alpha_m!} a_1^{\alpha_1} a_2^{\alpha_2} \cdots a_m^{\alpha_m}\).
• This generalization is particularly useful for understanding polynomial expansions in higher dimensions.

This powerful tool is employed extensively in combinatorics and analysis, providing insights into multisets and expansions of power series that are foundational in many areas of mathematics.