/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 1 Let \(z=f(x, y)\) be a function ... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 1

Let \(z=f(x, y)\) be a function of class \(C^{(1)}(G ; \mathbb{R})\). a) If \(\frac{\partial f}{\partial y}(x, y) \equiv 0\) in \(G\), can one assert that \(f\) is independent of \(y\) in \(G\) ? b) Under what condition on the domain \(G\) does the preceding question have an affirmative answer?

Short Answer

Expert verified
Answer: Yes, if the partial derivative of f(x,y) with respect to y is identically zero within the domain G and G is an open connected set, then f(x,y) can be considered independent of y within the domain G.

Step by step solution

01

Understand the condition\( \frac{\partial f}{\partial y}(x, y) \equiv 0\)

The given condition \(\frac{\partial f}{\partial y}(x, y) \equiv 0\) means that the partial derivative of the function \(f(x,y)\) with respect to \(y\) is identically zero in the domain \(G\). This means that any change in \(y\) does not affect the value of \(f(x,y)\). To put it another way, the function does not change with respect to \(y\).
02

Determine if the function is independent of \(y\)

Since the partial derivative of the function \(f(x,y)\) with respect to \(y\) is zero within the domain \(G\), we can assert that the function is indeed independent of \(y\) within this domain. However, the independence of the function on \(y\) may not hold true if there are any further constraints on the domain \(G\).
03

Analyze the domain \(G\) and any constraints for part b

In order for the function \(f(x,y)\) to be independent of \(y\) within the domain \(G\), the following condition must be met: The domain \(G\) must be an open connected set for any pair of points \((x_1,y_1)\) and \((x_2,y_2)\), implying that there exists a continuous path between these points that lies entirely within the domain \(G\). If this condition is met, it can be shown that \(\frac{\partial f}{\partial y}(x, y) \equiv 0\) implies that \(f(x,y)\) is indeed independent of \(y\) in the domain \(G\). To summarize: a) Yes, if \(\frac{\partial f}{\partial y}(x, y) \equiv 0\) in \(G\), it can be asserted that \(f\) is independent of y in \(G\), assuming \(G\) is an open connected set. b) The domain \(G\) must be an open connected set for any pair of points within it, in order for the preceding question to have an affirmative answer.

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Key Concepts

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

Independent Variable & Its Role in Functions
In mathematics, the term 'independent variable' refers to a variable whose variation does not depend on that of another. In the context of functions, an independent variable is one around which the function is structured or defined. For example, in the function \( z = f(x, y) \), \( x \) and \( y \) are often the independent variables as their values determine the output \( z \).

When we analyze partial derivatives, we are interested in how a function changes as one of the independent variables varies鈥攆or instance, using \( \frac{\partial f}{\partial y} \) to investigate changes in \( f \) when \( y \) changes, with \( x \) held constant. If the partial derivative with respect to a particular independent variable is zero throughout a domain, like the partial derivative of \( f \) concerning \( y \), \( \frac{\partial f}{\partial y}(x, y) \equiv 0 \), it suggests that within that domain, \( y \) does not influence the function's output. Therefore, \( f \) would be considered independent of \( y \) in that domain. This underscores the role of the independent variable, emphasizing its influence鈥攐r lack thereof鈥攐n the function's behavior.

Identifying independent and dependent variables is crucial for understanding how a system or function operates. It allows us to set expectations and predict outcomes based on given inputs in mathematical modeling and problem-solving scenarios.
Understanding Connected Domains
A connected domain is a critical concept in understanding functions, especially in multivariable calculus. A domain refers to the set of all possible input values for which the function is defined. When we say a domain is 'connected', it means any two points within that domain can be linked by a path that resides entirely inside the domain.

This is important for asserting properties about partial derivatives. For example, if we know \( \frac{\partial f}{\partial y}(x, y) \equiv 0 \) across a connected domain \( G \), it strongly indicates that \( f \) remains constant with changes in \( y \) across the entire path connecting any points in \( G \). This connection ensures that there are no isolated areas of varying function values due to \( y \), thus providing assurance of the function's consistent behavior relative to \( y \).

The concept of connectivity in a domain is useful in broader mathematical analyses and ensures that local conditions (like constant partial derivatives) hold globally within the domain, facilitating stronger conclusions in the analysis of functions.
Continuous Paths and Their Significance
Continuous paths within a domain are paths where you can travel from one point to another smoothly without interruptions. In mathematical terms, a continuous path is a continuous function \( \gamma: [a, b] \rightarrow G \), where \( G \) is our domain, and \( [a, b] \) is an interval.

In calculus, particularly when studying connected domains, continuous paths have significant implications. They ensure that if we know something about the function's behavior at one point, we can infer its behavior elsewhere, provided those points can be connected by a continuous path.

For the property \( \frac{\partial f}{\partial y}(x, y) \equiv 0 \) to universally imply that the function is independent of \( y \) throughout a connected domain, continuous paths play a key role. Continuous paths help substantiate the claim that the function does not change with \( y \) throughout \( G \) since any two points' behavior is directly linked by such paths.

Understanding continuous paths, therefore, allows us to apply local derivative conditions globally, ensuring that domains retain the same functional properties across all locations, which simplifies analysis and problem-solving in multivariable calculus.

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Most popular questions from this chapter

a) We shall regard two paths \(t \mapsto x_{1}(t)\) and \(t \mapsto x_{2}(t)\) as equivalent at the point \(x_{0} \in \mathbb{R}^{m}\) if \(x_{1}(0)=x_{2}(0)=x_{0}\) and \(\mathrm{d}\left(x_{1}(t), x_{2}(t)\right)=o(t)\) as \(t \rightarrow 0 .\) Verify that this relation is an equivalence relation, that is, it is reflexive, symmetric, and transitive. b) Verify that there is a one-to-one correspondence between vectors \(\mathbf{v} \in T \mathbb{R}_{x_{0}}^{m}\) and equivalence classes of smooth paths at the point \(x_{0}\). c) By identifying the tangent space \(T \mathbb{R}_{x 0}^{m}\) with the set of equivalence classes of smooth paths at the point \(x_{0} \in \mathbb{R}^{m}\), introduce the operations of addition and multiplication by a scalar for equivalence classes of paths. d) Determine whether the operations you have introduced depend on the coordinate system used in \(\mathbb{R}^{m}\).

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\()\).

Prove the following generalization of Rolle's theorem for functions of several variables. If the function \(f\) is continuous in a closed ball \(\bar{B}(0 ; r)\), equal to zero on the boundary of the ball, and differentiable in the open ball \(B(0 ; r)\), then at least one of the points of the open ball is a critical point of the function.

a) Verify that the tangent to a curve \(\Gamma: I \rightarrow \mathbb{R}^{m}\) is defined invariantly relative to the choice of coordinate system in \(\mathbb{R}^{m}\). b) Verify that the tangent plane to the graph \(S\) of a function \(y=f\left(x^{1}, \ldots, x^{m}\right)\) is defined invariantly relative to the choice of coordinate system in \(\mathbb{R}^{m}\). c) Suppose the set \(S \subset \mathbb{R}^{m} \times \mathbb{R}^{1}\) is the graph of a function \(y=f\left(x^{1}, \ldots, x^{m}\right)\) in coordinates \(\left(x^{1}, \ldots, x^{m}, y\right)\) in \(\mathbb{R}^{m} \times \mathbb{R}^{1}\) and the graph of a function \(\tilde{y}=\) \(\tilde{f}\left(\tilde{x}^{1}, \ldots, \tilde{x}^{m}\right)\) in coordinates \(\left(\tilde{x}^{1}, \ldots, \tilde{x}^{m}, \tilde{y}\right)\) in \(\mathbb{R}^{m} \times \mathbb{R}^{1}\). Verify that the tangent plane to \(S\) is invariant relative to a linear change of coordinates in \(\mathbb{R}^{m} \times \mathbb{R}^{1}\). d) Verify that the Laplacian \(\Delta f=\sum_{i=1}^{m} \frac{\partial^{2} f}{\partial x^{i^{2}}}(x)\) is defined invariantly relative to orthogonal coordinate transformations in \(\mathbb{R}^{m}\).

a) Give a direct proof of Morse's lemma for functions \(f: \mathbb{R} \rightarrow \mathbb{R}\). b) Determine whether Morse's lemma is applicable at the origin to the following functions: $$ \begin{gathered} f(x)=x^{3} ; \quad f(x)=x \sin \frac{1}{x} ; \quad f(x)=\mathrm{e}^{-1 / x^{2}} \sin ^{2} \frac{1}{x} \\ f(x, y)=x^{3}-3 x y^{2} ; \quad f(x, y)=x^{2} \end{gathered} $$ c) Show that nondegenerate critical points of a function \(f \in C^{(3)}\left(\mathbb{R}^{m} ; \mathbb{R}\right)\) are isolated: each of them has a neighborhood in which it is the only critical point of \(f\). d) Show that the number \(k\) of negative squares in the canonical representation of a function in the neighborhood of a nondegenerate critical point is independent of the reduction method, that is, independent of the coordinate system in which the function has canonical form. This number is called the index of the critical point.
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