boundary condition

Examples of boundary condition in the following topics:

• Logistic Equations and Population Grown

• Stokes' Theorem

  • Stokes' theorem relates the integral of the curl of a vector field over a surface to the line integral of the field around the boundary.
  • Stokes' theorem says that the integral of a differential form $\omega$ over the boundary of some orientable manifold $\Omega$ is equal to the integral of its exterior derivative $d\omega$ over the whole of $\Omega$, i.e.:
  • Given a vector field, the theorem relates the integral of the curl of the vector field over some surface to the line integral of the vector field around the boundary of the surface.
  • An illustration of the Kelvin–Stokes theorem, with surface $\Sigma$, its boundary $\partial$, and the "normal" vector $\mathbf{n}$.

• Volumes

  • Volumes of complicated shapes can be calculated using integral calculus if a formula exists for the shape's boundary.
  • Volume is the quantity of three-dimensional space enclosed by some closed boundary—for example, the space that a substance or shape occupies or contains.
  • The volumes of more complicated shapes can be calculated using integral calculus if a formula exists for the shape's boundary.

• The Divergence Theorem

  • Suppose $V$ is a subset of $R^n$ (in the case of $n=3$, $V$ represents a volume in 3D space) which is compact and has a piecewise smooth boundary $S$ (also indicated with $\partial V=S$).
  • The left side is a volume integral over the volume $V$; the right side is the surface integral over the boundary of the volume $V$.
  • It can not directly be used to calculate the flux through surfaces with boundaries, like those on the right.
  • (Surfaces are blue, boundaries are red.)

• Maximum and Minimum Values

  • Furthermore, a global maximum (or minimum) either must be a local maximum (or minimum) in the interior of the domain, or must lie on the boundary of the domain.
  • So a method of finding a global maximum (or minimum) is to look at all the local maxima (or minima) in the interior, and also look at the maxima (or minima) of the points on the boundary; and take the biggest (or smallest) one.

• Surfaces in Space

  • The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space $R^3$— for example, the surface of a ball.

• Curl and Divergence

  • The corresponding form of the fundamental theorem of calculus is Stokes' theorem, which relates the surface integral of the curl of a vector field to the line integral of the vector field around the boundary curve.
  • More rigorously, the divergence of a vector field $\mathbf{F}$ at a point $p$ is defined as the limit of the net flow of $\mathbf{F}$ across the smooth boundary of a three-dimensional region $V$ divided by the volume of $V$ as $V$ shrinks to $p$.
  • where $\left|V\right|$ is the volume of $V$, $S(V)$ is the boundary of $V$, and the integral is a surface integral with n being the outward unit normal to that surface.

• Parametric Surfaces and Surface Integrals

  • An illustration of the Kelvin–Stokes theorem, with surface $\Sigma$, its boundary $\partial$, and the "normal" vector $\mathbf{n}$.

• Models Using Differential Equations

  • Heat is being generated internally in the casing and being cooled at the boundary, providing a steady state temperature distribution.

• Green's Theorem

  • $D$ is a simple region with its boundary consisting of the curves $C_1$, $C_2$, $C_3$, $C_4$.