Nikiya Anton Bettey 2021: 09-22-2021-1003

Wednesday, September 22, 2021

09-22-2021-1003 - minimal surface

In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature(see definitions below).

The term "minimal surface" is used because these surfaces originally arose as surfaces that minimized total surface area subject to some constraint. Physical models of area-minimizing minimal surfaces can be made by dipping a wire frame into a soap solution, forming a soap film, which is a minimal surface whose boundary is the wire frame. However, the term is used for more general surfaces that may self-intersect or do not have constraints. For a given constraint there may also exist several minimal surfaces with different areas (for example, see minimal surface of revolution): the standard definitions only relate to a local optimum, not a global optimum.

A helicoid minimal surface formed by a soap film on a helical frame

https://en.wikipedia.org/wiki/Minimal_surface

Minimal surfaces

Associate family Bour's Catalan's Catenoid Chen–Gackstatter Costa's Enneper Gyroid Helicoid Henneberg k-noid Lidinoid Neovius Richmond Riemann's Saddle tower Scherk Schwarz Triply periodic

https://en.wikipedia.org/wiki/Catalan%27s_minimal_surface

Subcategories

This category has the following 9 subcategories, out of 9 total.

Pages in category "Surfaces"

The following 124 pages are in this category, out of 124 total. This list may not reflect recent changes (learn more).

A

B

C

D

E

F

G

H

I

J

Jacob's ladder surface

K

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M

N

O

Orientability

P

R

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T

U

W

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Zoll surface

https://en.wikipedia.org/wiki/Category:Surfaces

In mathematics, Plateau's problem is to show the existence of a minimal surface with a given boundary, a problem raised by Joseph-Louis Lagrange in 1760. However, it is named after Joseph Plateau who experimented with soap films. The problem is considered part of the calculus of variations. The existence and regularity problems are part of geometric measure theory.

https://en.wikipedia.org/wiki/Plateau%27s_problem

In mathematics and mathematical physics, potential theory is the study of harmonic functions.

https://en.wikipedia.org/wiki/Potential_theory

The calculus of variations is a field of mathematical analysis that uses variations, which are small changes in functionsand functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers.^[a]Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations.

https://en.wikipedia.org/wiki/Calculus_of_variations

In the field of differential geometry in mathematics, mean curvature flow is an example of a geometric flow of hypersurfaces in a Riemannian manifold (for example, smooth surfaces in 3-dimensional Euclidean space). Intuitively, a family of surfaces evolves under mean curvature flow if the normal component of the velocity of which a point on the surface moves is given by the mean curvature of the surface. For example, a round sphere evolves under mean curvature flow by shrinking inward uniformly (since the mean curvature vector of a sphere points inward). Except in special cases, the mean curvature flow develops singularities.

Under the constraint that volume enclosed is constant, this is called surface tension flow.

It is a parabolic partial differential equation, and can be interpreted as "smoothing".

https://en.wikipedia.org/wiki/Mean_curvature_flow

In physics, the Young–Laplace equation (/ləˈplɑːs/) is a nonlinear partial differential equation that describes the capillary pressuredifference sustained across the interface between two static fluids, such as water and air, due to the phenomenon of surface tension or wall tension, although use of the latter is only applicable if assuming that the wall is very thin. The Young–Laplace equation relates the pressure difference to the shape of the surface or wall and it is fundamentally important in the study of static capillary surfaces. It is a statement of normal stress balance for static fluids meeting at an interface, where the interface is treated as a surface (zero thickness):

{\begin{aligned}\Delta p&=-\gamma \nabla \cdot {\hat {n}}\\&=-\gamma H_{f}\\&=-\gamma \left({\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}\right)\end{aligned}}

where $\Delta p$ is the Laplace pressure, the pressure difference across the fluid interface (the exterior pressure minus the interior pressure), $\gamma$ is the surface tension (or wall tension), ${\hat {n}}$ is the unit normal pointing out of the surface, $H_{f}$ is the mean curvature, and $R_{1}$ and $R_{2}$ are the principal radii of curvature. Note that only normal stress is considered, this is because it has been shown^[1] that a static interface is possible only in the absence of tangential stress.

The equation is named after Thomas Young, who developed the qualitative theory of surface tension in 1805, and Pierre-Simon Laplace who completed the mathematical description in the following year. It is sometimes also called the Young–Laplace–Gauss equation, as Carl Friedrich Gauss unified the work of Young and Laplace in 1830, deriving both the differential equation and boundary conditions using Johann Bernoulli's virtual work principles.^[2]

https://en.wikipedia.org/wiki/Young–Laplace_equation

In the mathematical study of the differential geometry of surfaces, a tangent developable is a particular kind of developable surface obtained from a curve in Euclidean space as the surface swept out by the tangent lines to the curve. Such a surface is also the envelope of the tangent planes to the curve.

The tangent developable of a helix

https://en.wikipedia.org/wiki/Tangent_developable

In geometry, an envelope of a planar family of curves is a curve that is tangent to each member of the family at some point, and these points of tangency together form the whole envelope. Classically, a point on the envelope can be thought of as the intersection of two "infinitesimally adjacent" curves, meaning the limitof intersections of nearby curves. This idea can be generalized to an envelope of surfaces in space, and so on to higher dimensions.

https://en.wikipedia.org/wiki/Envelope_(mathematics)

In mathematics, an implicit curve is a plane curve defined by an implicit equation relating two coordinate variables, commonly x and y. For example, the unit circle is defined by the implicit equation $x^{2}+y^{2}=1$ . In general, every implicit curve is defined by an equation of the form

F(x,y)=0

for some function F of two variables. Hence an implicit curve can be considered as the set of zeros of a functionof two variables. Implicit means that the equation is not expressed as a solution for either x in terms of y or vice versa.

If $F(x,y)$ is a polynomial in two variables, the corresponding curve is called an algebraic curve, and specific methods are available for studying it.

Plane curves can be represented in Cartesian coordinates (x, y coordinates) by any of three methods, one of which is the implicit equation given above. The graph of a function is usually described by an equation $y=f(x)$ in which the functional form is explicitly stated; this is called an explicit representation. The third essential description of a curve is the parametric one, where the x- and y-coordinates of curve points are represented by two functions $x (t), y (t)$ both of whose functional forms are explicitly stated, and which are dependent on a common parameter $t.$

Examples of implicit curves include:

a line: $x+2y-3=0,$
a circle: $x^{2}+y^{2}-4=0,$
the semicubical parabola: $x^{3}-y^{2}=0,$
Cassini ovals $(x^{2}+y^{2})^{2}-2c^{2}(x^{2}-y^{2})-(a^{4}-c^{4})=0$ (see diagram),
$\sin(x+y)-\cos(xy)+1=0$ (see diagram).

The first four examples are algebraic curves, but the last one is not algebraic. The first three examples possess simple parametric representations, which is not true for the fourth and fifth examples. The fifth example shows the possibly complicated geometric structure of an implicit curve.

The implicit function theorem describes conditions under which an equation $F(x,y)=0$ can be solved implicitly for x and/or y – that is, under which one can validly write $x=g(y)$ or $y=f(x)$ . This theorem is the key for the computation of essential geometric features of the curve: tangents, normals, and curvature. In practice implicit curves have an essential drawback: their visualization is difficult. But there are computer programs enabling one to display an implicit curve. Special properties of implicit curves make them essential tools in geometry and computer graphics.

An implicit curve with an equation $F(x,y)=0$ can be considered as the level curve of level 0 of the surface $z=F(x,y)$ (see third diagram).

https://en.wikipedia.org/wiki/Implicit_curve

In mathematics, more specifically in multivariable calculus, the implicit function theorem^[a] is a tool that allows relations to be converted to functions of several real variables. It does so by representing the relation as the graph of a function. There may not be a single function whose graph can represent the entire relation, but there may be such a function on a restriction of the domain of the relation. The implicit function theorem gives a sufficient condition to ensure that there is such a function.

More precisely, given a system of $m$ equations $f i (x 1, ..., x n, y 1, ..., y m) = 0, i = 1, ..., m$ (often abbreviated into $F (x, y) = 0$ ), the theorem states that, under a mild condition on the partial derivatives (with respect to the $y i$ s) at a point, the $m$ variables $y i$ are differentiable functions of the $x j$ in some neighborhood of the point. As these functions can generally not be expressed in closed form, they are implicitly defined by the equations, and this motivated the name of the theorem.^[1]

In other words, under a mild condition on the partial derivatives, the set of zeros of a system of equations is locally the graph of a function.

https://en.wikipedia.org/wiki/Implicit_function_theorem

Constant rank theorem[edit]

The inverse function theorem (and the implicit function theorem) can be seen as a special case of the constant rank theorem, which states that a smooth map with constant rank near a point can be put in a particular normal form near that point.^[11] Specifically, if $F:M\to N$ has constant rank near a point $p\in M\!$ , then there are open neighborhoods $U$ of $p$ and $V$ of $F(p)\!$ and there are diffeomorphisms $u:T_{p}M\to U\!$ and $v:T_{F(p)}N\to V\!$ such that $F(U)\subseteq V\!$ and such that the derivative $dF_{p}:T_{p}M\to T_{F(p)}N\!$ is equal to $v^{-1}\circ F\circ u\!$ . That is, $F$ "looks like" its derivative near $p$ . The set of points $p\in M$ such that the rank is constant in a neighbourhood of $p$ is an open dense subset of $M$ ; this is a consequence of semicontinuity of the rank function. Thus the constant rank theorem applies to a generic point of the domain.

When the derivative of $F$ is injective (resp. surjective) at a point $p$ , it is also injective (resp. surjective) in a neighborhood of $p$ , and hence the rank of $F$ is constant on that neighborhood, and the constant rank theorem applies.

Holomorphic functions[edit]

If a holomorphic function $F$ is defined from an open set $U$ of $\mathbb {C} ^{n}\!$ into $\mathbb {C} ^{n}\!$ , and the Jacobian matrix of complex derivatives is invertible at a point $p$ , then $F$ is an invertible function near $p$ . This follows immediately from the real multivariable version of the theorem. One can also show that the inverse function is again holomorphic.^[12]

https://en.wikipedia.org/wiki/Inverse_function_theorem#Constant_rank_theorem

In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are smooth.

https://en.wikipedia.org/wiki/Diffeomorphism

In mathematics, a surjective function (also known as surjection, or onto function) is a function $f$ that maps an element $x$ to every element $y$ ; that is, for every $y$ , there is an $x$ such that $f (x) = y$ . In other words, every element of the function's codomain is the image of at least one element of its domain.^[1]^[2]^[3] It is not required that $x$ be unique; the function $f$ may map one or more elements of $X$ to the same element of $Y$ .

https://en.wikipedia.org/wiki/Surjective_function

In mathematics, a constant function is a function whose (output) value is the same for every input value.^[1]^[2]^[3] For example, the function $y (x) = 4$ is a constant function because the value of $y (x)$ is 4 regardless of the input value $x$ (see image).

https://en.wikipedia.org/wiki/Constant_function

In mathematics, a continuous function is a function that does not have any abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its output can be assured by restricting to sufficiently small changes in its input. If not continuous, a function is said to be discontinuous. Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, during which attempts such as the epsilon–delta definition were made to formalize it.

Continuity of functions is one of the core concepts of topology, which is treated in full generality below. The introductory portion of this article focuses on the special case where the inputs and outputs of functions are real numbers. A stronger form of continuity is uniform continuity. In addition, this article discusses the definition for the more general case of functions between two metric spaces. In order theory, especially in domain theory, one considers a notion of continuity known as Scott continuity. Other forms of continuity do exist but they are not discussed in this article.

As an example, the function $H (t)$ denoting the height of a growing flower at time $t$ would be considered continuous. In contrast, the function $M (t)$ denoting the amount of money in a bank account at time $t$ would be considered discontinuous, since it "jumps" at each point in time when money is deposited or withdrawn.

https://en.wikipedia.org/wiki/Continuous_function#Continuous_functions_between_topological_spaces

In mathematics, a function f is uniformly continuous if, roughly speaking, it is possible to guarantee that f(x) and f(y) be as close to each other as we please by requiring only that x and y be sufficiently close to each other; unlike ordinary continuity, where the maximum distance between f(x) and f(y) may depend on x and ythemselves.

Continuous functions can fail to be uniformly continuous if they are unbounded on a finite domain, such as $f(x)={\tfrac {1}{x}}$ on (0,1), or if their slopes become unbounded on an infinite domain, such as $f(x)=x^{2}$ on the real line. However, any Lipschitz map between metric spaces is uniformly continuous, in particular any isometry (distance-preserving map).

Although ordinary continuity can be defined for functions between general topological spaces, defining uniform continuity requires more structure. The concept relies on comparing the sizes of neighbourhoods of distinct points, so it requires a metric space, or more generally a uniform space.

https://en.wikipedia.org/wiki/Uniform_continuity

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