Integral Calculus - Surface Integral 1

Rabu, 29 Juli 2020

Surface Integral 1

 

 

• If the surface S is given in parametric by the vector

      , then the latter formula can 
       be written as

     

  Where (u, v) ranges over some domain D(u, v) of 

  the uv-plane.

 

• Divergence Theorem

,

   Where

      

   is a vector field whose components P, Q, and R 

   have continuous partial derivatives,

      

   is the divergence of, also denoted. 

   The symbolindicates that the surface integral 

   is taken over a closed surface.

 

• Divergence Theorem in Coordinate Form

  .

 

• Stoke’s Theorem

   ,

    where

       

    is a vector field whose components P, Q, and 

    R have continuous partial derivatives,

       

    is the curl of, also denoted.

    The symbolindicates that the line integral is 

    taken over a closed curve.

 

• Surface Area

      

• If the surface S is parameteri by the vector

   ,

   Then the surface area is

   ,

   Where D(u, v) is the domain where the surface

   is defined.

 

• If the surface S is given explicitly by the function z(x, y), then the surface area is

   ,

   where D(x, y) is the projection of the surface S onto 

   the xy-plane.

 

• Mass of a Surface

      

   where μ(x, y, z) is the mass per unit area (density function).

 

• Center of Mass of a Shell

      

  are the first moment about the coordinate plane 

  x = 0, y = 0, z = 0, respective. μ(x, y, z) is the density 

  function.

 

• Moments of Inertia about the xy-plane (or z = 0), yz-plane (x = 0), and xz-plane (y = 0)

      

• Moments of Inertia about the x-axis, y-axis, and z-axis

      

• Volume of a Solid Bounded by a Closed Surface

      

• Gravitational Force

      

   where m is a mass at a point 〈x₀, y₀, z₀〉 outside the 

   surface,

       

   μ(x, y, z) is the density function, and G is gravitational 

   constant.

 

• Pressure Force

      

   where the pressureacts on the surface S 

   given by the position vector.

 

• Fluid Flux (across the surface S)

     

   whereis the fluid velocity.

 

• Mass Flux (across the surface S)

     

   whereis the vector field, ρ is the fluid density.

 

• Surface Charge

     

   where σ(x, y) is the surface charge density.

 

• Gauss’ Law

   The electric flux through any closed surface is 

   proportional to the charge Q enclosed by the surface

      ,

   where

   Φ is the electric flux,

   is the magnitude of the electric field strength,

   is permittivity of free space.

 

Sumber

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