Surface Integral: Formula & Application

Surface Integral is the generalization of several multiple integrals over the integration surfaces. It is used to add a bunch of values with the points over the surface. 

• The surface integral is calculated just like we calculate the surface area using the double integral method.
• This integral is considered an analogue of the line integral.
• The method allows one to integrate over a two-dimensional surface, and as a result, we can think of it as a double integral.
• In this integral, the element of the area being a vector is normal to the surface.
• Closed surface and open surface are used to determine the positive direction of surface integrals.
• Gravitational forces and centre of mass are physical applications of surface integral.

Key Terms: Integrals, Surface Integral, Surface Integral Formula, Surface Integral of Scalar Function, Surface Integral of Vector Function, Line Integral, Integration

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Surface Integral Definition

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A surface integral is an integral that can handle the integration of objects at higher dimensions. It is used in determing the higher versions of the fundamental theorem of calculus.

• The integrals is used to solve problem over surface either in the scalar field or the vector field. 
• If we want to calculate a surface integral over the surface S then we should parameterize S.
• Scalar field will return position of function as the final scalar value.
• On the other hand, vector field will return function as the vector value.
• In this method the value of the function is stick inside the integral.
• Surface integrals is used in the field of physics to solve problem of electromagnetism.
• We can represent a parameterized surface as;

r(u,v) = (x(u,v),y(u,v),z(u,v))

• where u and v represent parameter domain
• r represents parameter region

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Surface Integral Formula

The Surface integral is divided into two types namely scalar valued function and vector valued function which are as follows:

Surface Integral of Scalar Field

The surface integral of scalar field can be explained with the derivation. Let’s consider a surface that is S & its functions that will be represented with f (x, y, z) 

• Suppose that S is denoted by the position vector,

r(u, v)= x(u, v)j + z(u, v)k

• Hence, the Surface Integral will be:

∫sf(x,y,z)dS = ∫∫D (u, v) fx (u,v), y(u,v), z(u, v)

∂r/∂u × ∂r/∂v|

• Whereas, the range of coordinates in the domain of the UV-plane is (u, v)
• All the cross products are perpendicular to the surface at a point 
• (u, v) are ∂r/∂u × ∂r/∂v|
• Hence, the partial derivatives are ∂r/∂u × ∂r/∂v|
• Thus, the absolute value which is ∂r/∂u × ∂r/∂v| can be referred to as the area aspect.

                                                                                                                    Surface Integral of Scalar Field

Surface Integral of Vector Field

For calculating, the surface integral of Vector fields we should first, consider a vector field having a surface S and the functions are represented as F(x, y, z)

• We can define it continuously with the position of the vector;

r(u, v)= x(u, v)j + z(u, v)k

• Suppose, that n(x, y, z) is a normal vector unit to the surface at the point (x, y, z)
• The surface S, is smooth & it has a continuous function n(x, y, z) 
• So, the possibilities are two 

N(x, y, z) & n(x, y, z)

• If the surface (S) is oriented outward;

∫∫sF(x,y,z).dS = ∫∫sF(x,y,z). ndS = ∫∫D u,vFxu,v,y,zu,v. |∂r∂u × ∂r∂v| dudv

• If the surface (S) is oriented inward;

∫∫sF(x,y,z).dS = ∫∫sF(x,y,z). ndS = ∫∫D u,vFxu,v,y,z. |∂r∂u × ∂r∂v| dudv

• Thus, dS = ndS= the Vector element of surface S.

Surface Integral of Vector Field 

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Applications of Surface Integral 

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The surface integral is vastly used in different fields like engineering and science. It has a large number of applications, which are as follows:

• The surface integral can be used to find out the centre of mass.
• It is used to find the rotational inertia of a shell.
• The method helps find the mass of a shell.
• It can be used for calculating the electric charge of a surface.
• The surface integral is used to solve the problem of the electric field. 
• The concept is used for determining the gravitational force & pressure can also be determined with surface integrals.

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Things to Remember

• The surface integral is used when you want to add up a bunch of values that are associated with some points over a surface.
• In simple words, the integral is the generalization of so many multiple integrals for integration over the surface.
• The scalar field involves the integration of Riemannian volume form on the parameterized surface.
• Vector field is a special case of integrating 2-forms.
• We can find the surface integral of a closed area, like a shape that is closed from all sides.