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  2. Implicit function theorem - Wikipedia

    en.wikipedia.org/wiki/Implicit_function_theorem

    The unit circle can be specified as the level curve f(x, y) = 1 of the function f(x, y) = x 2 + y 2.Around point A, y can be expressed as a function y(x).In this example this function can be written explicitly as () =; in many cases no such explicit expression exists, but one can still refer to the implicit function y(x).

  3. Nash embedding theorems - Wikipedia

    en.wikipedia.org/wiki/Nash_embedding_theorems

    The Nash embedding theorem is a global theorem in the sense that the whole manifold is embedded into R n. A local embedding theorem is much simpler and can be proved using the implicit function theorem of advanced calculus in a coordinate neighborhood of the manifold. The proof of the global embedding theorem relies on Nash's implicit function ...

  4. Implicit function - Wikipedia

    en.wikipedia.org/wiki/Implicit_function

    An implicit function is a function that is defined by an implicit equation, that relates one of the variables, considered as the value of the function, with the others considered as the arguments. [ 1 ] : 204–206 For example, the equation x 2 + y 2 − 1 = 0 {\displaystyle x^{2}+y^{2}-1=0} of the unit circle defines y as an implicit function ...

  5. Lyapunov–Schmidt reduction - Wikipedia

    en.wikipedia.org/wiki/Lyapunov–Schmidt_reduction

    For the case when the linear operator (,) is invertible, the implicit function theorem assures that there exists a solution () satisfying the equation ((),) = at least locally close to . In the opposite case, when the linear operator f x ( x , λ ) {\displaystyle f_{x}(x,\lambda )} is non-invertible, the Lyapunov–Schmidt reduction can be ...

  6. Steven G. Krantz - Wikipedia

    en.wikipedia.org/wiki/Steven_G._Krantz

    Krantz's monographs include Function Theory of Several Complex Variables, Complex Analysis: The Geometric Viewpoint, A Primer of Real Analytic Functions (joint with Harold R. Parks), The Implicit Function Theorem (joint with Harold Parks), Geometric Integration Theory (joint with Harold Parks), and The Geometry of Complex Domains (joint with ...

  7. Inverse function theorem - Wikipedia

    en.wikipedia.org/wiki/Inverse_function_theorem

    (Note the proof is quite similar to the proof of the implicit function theorem and, in fact, the implicit function theorem can be also used instead.) More generally, the theorem shows that if a smooth map f : P → E {\displaystyle f:P\to E} is transversal to a submanifold M ⊂ E {\displaystyle M\subset E} , then the pre-image f − 1 ( M ...

  8. 9 Items You Should Actually Store In The Freezer, According ...

    www.aol.com/9-items-actually-store-freezer...

    Simply place your bottle of vodka, rum, tequila—any spirit with more than 40% alcohol by volume (ABV) or 80 proof—in the freezer. A properly-sealed bottle of high-proof alcohol should last for ...

  9. Differential geometry of surfaces - Wikipedia

    en.wikipedia.org/wiki/Differential_geometry_of...

    A major theorem, often called the fundamental theorem of the differential geometry of surfaces, asserts that whenever two objects satisfy the Gauss-Codazzi constraints, they will arise as the first and second fundamental forms of a regular surface. Using the first fundamental form, it is possible to define new objects on a regular surface.