Integral Calculus - Surface Integral

Selasa, 28 Juli 2020

Surface Integral

 

 

Scalar functions: f(x, y, z), z(x, y)

Position vectors:

Unit vectors:

Surface: S

Vector field:

Divergence of a vector field:

Curl of a vector field:

Vector element of a surface:

Normal to surface:

Surface area: A

Mass of a surface: m

Density: μ(x, y, z)

Coordinates of center of mass:

First moments of inertia: M_xy, M_yz, M_xz

Moments of inertia: I_xy, I_yz, I_xz, I_x, I_y, I_z

Volume of a solid: V

Force:

Gravitational constant: G

Fluid velocity:

Fluid density: ρ

Pressure:

Mass flux, electri flux: Φ

Surface charge: Q

Charge density: σ(x, y

Magnitude of the electric field:

 

• Surface Integral of a Scalar Function

   Let surface S be given by the position vector

,

   where (u, v) ranges over some domain D(u, v) of the 

    uv-plane.

   The surface integral of a scalar function f(x, y, z) over 

   the surface S is defined as

      

   where the partial derivativesandare 

   given by

      

• If the surface S is given by the equation z = z(x, y) where z(x, y) is a differentiable function in the domain D(x, y), then

     

 

• Surface Integral of the Vector Field  over the Surface S

   a. If S is oriented outward, then

   

   

  b. If S is oriented inward, then

     

  

 

is called the vector element of the surface. 

    Dot means the scalar product of the appropriate 

    vevtors. The partial derivativesandare 

    given by

        

• If the surface S is given by the equation z = z(x, y), where z(x, y), is a differentiable function in the domain D(x, y), then

   a. If S is oriented upward, i.e, the k-th component of 

       the normal vector is positive, then

   

  

 

  b. If S is oriented downward, i.e, the k-th component of 

      the normal vector is negative, then

     

  

 

where P(x, y, z), Q(x, y, z), R(x, y, z) are the component of the vector field.

cos α, cos β , cos γ are the angles between the outter unit normal vector and the x-axis,y-axis, and z-axis, respectively.

 

 

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