8.1 Sard's Theorem and critical values
3 min read•august 9, 2024
is a game-changer in differential topology. It tells us that for smooth functions, critical values are rare, allowing us to focus on regular values. This concept is crucial for understanding the behavior of functions between manifolds.
The theorem has far-reaching implications, from proving the existence of regular values to supporting advanced topics like . It's a powerful tool that helps us navigate the complex landscape of smooth maps and their properties.
Critical Points and Values
Understanding Critical Points and Values
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- Critical points occur where the derivative or of a function vanishes or is not defined
- Critical values result from mapping critical points through the function
- Regular values comprise all points in the codomain that are not critical values
- represents the best linear approximation of a function near a given point
- Rank of a matrix determines the dimension of its image, crucial for identifying critical points
Applications of Critical Points in Analysis
- Critical points help identify local extrema, saddle points, and inflection points in functions
- states that the preimage of a regular value is a submanifold
- Morse theory uses critical points to study the topology of manifolds
- vanishing indicates potential critical points in multivariable calculus
- Rank deficiency in the Jacobian matrix signals the presence of critical points
Examples and Visualizations
- For , critical points occur at (local extrema)
- In two dimensions, has a at (global minimum)
- Jacobian matrix for is
- Rank of the Jacobian drops to 1 at , indicating a critical point
- Regular values of include all positive real numbers, while 0 is the only
Sard's Theorem
Fundamentals of Sard's Theorem
- Sard's Theorem states that the set of critical values of a has
- Smooth maps refer to infinitely differentiable functions between manifolds
- Measure zero sets have "negligible" size, often visualized as dust-like in the codomain
- Theorem applies to functions between finite-dimensional spaces or manifolds
- Generalizes to infinite dimensions under certain conditions ()
Implications and Applications of Sard's Theorem
- Ensures that "most" points in the codomain are regular values
- Facilitates the use of in differential topology
- Proves the existence of regular values for smooth functions
- Supports the study of singular points in algebraic geometry
- Contributes to the development of Morse theory and
Examples and Extensions
- For , , the critical value set has measure zero
- Sard's Theorem applies to the Gauss map on surfaces, crucial in differential geometry
- In complex analysis, the set of critical values of a holomorphic function is countable
- extends Sard's result to certain infinite-dimensional spaces
- builds upon Sard's Theorem for jet spaces
Key Terms to Review (24)
Continuity: Continuity refers to the property of a function or mapping that preserves the closeness of points, ensuring that small changes in input lead to small changes in output. This concept is crucial in understanding how functions behave, especially when analyzing their inverses, applying derivatives, or investigating the structure of manifolds.
Critical Point: A critical point is a point on a differentiable function where its derivative is either zero or undefined, indicating a potential local maximum, local minimum, or saddle point. Understanding critical points is essential in analyzing smooth maps, immersions, and their properties, as well as in applying various theorems related to differential topology.
Critical Value: A critical value refers to a point in the domain of a function where its derivative is either zero or undefined, indicating potential maxima, minima, or points of inflection. These values are essential in understanding the behavior of functions, especially when analyzing smooth mappings between manifolds, as they help identify where the function fails to be a submersion. Additionally, critical values play a significant role in Sard's Theorem, which deals with the measure of sets of critical values and their implications on the image of functions.
Degenerate critical point: A degenerate critical point is a point in the domain of a differentiable function where the gradient is zero, and the Hessian matrix is not invertible, indicating that the behavior of the function around this point is not typical. This means that instead of a standard local minimum or maximum, the nature of this point can be ambiguous, leading to various possibilities such as saddle points or flat regions in the function's surface. Understanding degenerate critical points is essential when applying Sard's Theorem to identify critical values and analyze the topology of level sets.
Differentiable Manifold: A differentiable manifold is a topological space that locally resembles Euclidean space and has a consistent way to differentiate functions defined on it. This structure allows for the application of calculus in higher dimensions, enabling us to analyze smooth curves and surfaces within a broader context.
F: m → n: The notation f: m → n describes a function f that maps elements from a set m (the domain) to elements in another set n (the codomain). This notation is crucial for understanding how functions relate to the concepts of continuity, differentiability, and critical values, as it illustrates how inputs from one space correspond to outputs in another.
F'(x): The notation f'(x) represents the derivative of a function f at the point x, indicating the rate at which the function's value changes with respect to changes in x. This concept is crucial for understanding how functions behave locally, especially in identifying critical points where the function may have maxima, minima, or points of inflection. The derivative provides essential information about the slope of the tangent line to the function's graph at any given point.
Jacobian: The Jacobian is a matrix that represents the rates of change of a set of functions with respect to a set of variables. It captures how a function maps changes in input space to changes in output space and is fundamental in understanding differentiability, especially in higher dimensions. The Jacobian plays a crucial role in the analysis of critical points, transformation of variables, and computation of degrees in topology.
Jacobian Determinant: The Jacobian determinant is a scalar value that represents the rate of change of a vector-valued function with respect to its input variables. It plays a crucial role in understanding how functions behave under transformations, especially in relation to critical points and mapping properties. By examining the Jacobian determinant, one can determine whether a function is locally invertible and analyze the behavior of maps, which is essential in studying critical values and degrees of mappings.
Jacobian Matrix: The Jacobian matrix is a matrix of first-order partial derivatives of a vector-valued function. It provides crucial information about the local behavior of functions in multivariable calculus, particularly when considering transformations and their effects on space. The Jacobian plays a vital role in understanding how changes in input variables affect output variables, making it essential for analyzing concepts like the inverse function, differentials, and critical values.
John Milnor: John Milnor is a prominent American mathematician known for his significant contributions to differential topology, particularly in the areas of manifold theory, Morse theory, and the topology of high-dimensional spaces. His work has fundamentally shaped the field and has broad implications for various topics within topology, including submersions, critical values, and cohomology groups.
Measure zero: A set is considered to have measure zero if, intuitively speaking, it is so small that it can be covered by a collection of intervals or sets whose total length can be made arbitrarily small. This concept is essential in understanding the properties of critical values and the application of Sard's theorem, which relates to the behavior of smooth functions and their critical points. Measure zero sets play a significant role in analysis and topology, particularly when examining properties of functions and differentiable maps.
Morse theory: Morse theory is a branch of mathematics that studies the topology of manifolds using smooth functions, particularly focusing on the critical points of these functions and their implications for the manifold's structure. By analyzing how these critical points behave under variations of the function, Morse theory connects the geometry of the manifold with its topology, providing deep insights into the shape and features of the space.
Non-degenerate critical point: A non-degenerate critical point of a smooth function is a point where the gradient of the function is zero, and the Hessian matrix at that point is invertible. This means that at a non-degenerate critical point, the second derivative test can be applied, leading to definitive conclusions about the nature of the critical point, whether it is a local minimum, local maximum, or a saddle point. Understanding these points is crucial in analyzing the behavior of functions and plays a significant role in various mathematical theories.
Rank Condition: The rank condition refers to a specific requirement related to the behavior of differentiable functions and the dimensionality of their images. It often indicates that the differential of a function has full rank at a point, which is crucial for applying various results in differential topology, particularly regarding the existence of local inverse functions and understanding critical values in mappings.
Regular Values Theorem: The Regular Values Theorem states that if a smooth map between manifolds has certain properties, then the preimage of regular values under that map is a submanifold of the domain manifold. This concept is essential in understanding how critical points and values behave in differential topology, particularly in relation to Sard's Theorem, which deals with the relationship between critical values and regular values.
Rufus Sard: Rufus Sard was an American mathematician known for his contributions to differential topology, particularly in the formulation of Sard's Theorem. This theorem is crucial in understanding the behavior of smooth functions and identifies the critical values of these functions, which are essential for understanding the structure of manifolds and mappings between them.
Sard's Theorem: Sard's Theorem states that the set of critical values of a smooth map between manifolds has measure zero. This means that, when mapping from one manifold to another, most points in the target manifold are regular values, which helps understand how the smooth map behaves. The implications of this theorem stretch into various areas, such as the properties of submersions and regular values, influencing the topology and geometry involved in differentiable maps.
Singularity Theory: Singularity theory studies the behavior of mathematical functions and mappings near points where they fail to be well-behaved, called singularities. These points can cause sudden changes or disruptions in the behavior of functions, which is crucial in understanding their overall structure and properties. The insights from singularity theory are used in various applications, including critical points in differential topology, leading to important results like Sard's Theorem, which deals with the properties of critical values and their significance in understanding smooth mappings.
Smale-Sard Theorem: The Smale-Sard Theorem states that for a smooth map between differentiable manifolds, the set of critical values (the images of critical points) has measure zero. This result highlights the rarity of critical values and provides a significant insight into the behavior of smooth functions, indicating that most values in the target manifold are not critical values.
Smooth function: A smooth function is a function that has continuous derivatives of all orders. This property ensures that the function behaves nicely and can be differentiated repeatedly without encountering any abrupt changes or discontinuities. The concept of smoothness is crucial when discussing various mathematical results and theorems, as it allows for a deeper understanding of how functions interact with their environments in a differentiable context.
Thom's Transversality Theorem: Thom's Transversality Theorem is a fundamental result in differential topology that states that, under certain conditions, a smooth map between manifolds can be made transverse to a submanifold by a small perturbation of the map. This theorem has profound implications, especially in understanding the structure of critical values and critical points, and it lays the groundwork for several important applications in differential topology.
Transversality: Transversality is a concept in differential topology that describes the condition where two submanifolds intersect in a way that is 'nice' or 'generic', meaning they meet at a finite number of points and the tangent spaces at those points span the ambient space. This idea is essential for understanding the behavior of functions and their critical values, as well as the relationships between different geometric objects.
Whitney's Theorem: Whitney's Theorem states that for a smooth manifold, the set of immersions into Euclidean space can be characterized by certain properties related to the dimensions of the manifold and the ambient space. It highlights the relationship between immersions, transversality, and the critical values of smooth functions, establishing key connections in differential topology that are essential for understanding the behavior of mappings between manifolds.