Integral Calculus - Line Integral

Sabtu, 25 Juli 2020

Line Integral

 

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

Scalar potential: u(x, y, z)

Curves: C, C₁, C₂

Limits of integrations: a, b, α, β

Parameters: t, s

Polar coordinates: r, θ

Vector field:

Position vectors:

Unit vectors:

Area of region: S

Length of a curve: L

Mass of a wire: m

Density: ρ(x, y, z), ρ(x, y)

Coordinates of center of mass:

First moments: M_xy, M_yz, M_xz,

Moments of inertia: I_x, I_y, I_z

Volume of a solid: V

Work: W

Magnetic field:

Current: I

Electromotive force: ε

Magnetic flux: ψ

 

• Line Integral of a Scalar Function

  Let a curve C be given by the vector function, 

  0 ≤ s ≤ S, and scalar function F is defined over the 

  curve C.

  Then

     

  Where ds is the arc length differential.

 

 

Figure

 

• If the smooth curve C is parametrized by, α ≤ t ≤ β, then

     

• If C is a smooth curve in the xy-plane given by the equation y = f(x), a ≤ x ≤ b, then

     

• Line Integral of Scalar Function in Polar Coordinater

     

  Where the curve C is defined by the polar function r(θ)

 

• Line Integral of Vector Field

  Let a curve C be defined by the vector function
  , 0 ≤ x ≤ S. Then

    

  Is the unit vector of the tangent line to this curve.

 

Figure

 

  Let a vector fieldis defined over the curve C. 

  Then the line of the vector fieldalong the curve C 

  is

    

• Properties of Integral of Vector Field

     

   where – C denote the curve with the opposite.

 

   where C is the union of the curve C₁ and C₂.

 

• If the curve C is parameterized by
  , α ≤ t ≤ β, then

    

• If C lies in the xy-plane and given by the equation y = f(x ), then  

   

 

• Green’s Theorem

   

whereis a continuous vector function with continuous first partial derivativesin a some domain R, which is bouded by a closed, piecewise smooth curve C.

 

Sumber

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