Surface Integral: Overview, Questions, Preparation

A surface integral comes handy while adding values to a set of points that lie on a given surface. It is also referred to as double integral sometimes. The surface integrals can be calculated in both vector and scalar fields. Accordingly, they are referred to as the surface integral of a vector or surface integral of a scalar, respectively. 

Formula for Computing Surface Integral 

The formula for computing surface integral depends on whether the function is scalar or vector. Its calculation is done in the same way as the calculation of double integral is done. However, in this case, the function that falls in the integrals is not calculated. In the vector field, the orientation plays a huge role in determining the values of the surface integrals as it depends whether the surface is oriented in the inward or outward direction. The formula for these respective fields are given below:

A Scalar Field’s Surface Integral

When the function or field is scalar, then the surface integral can be defined as below:

∬S f(x,y,z)dS = ∬D(u,v) f[x(u,v),y(u,v),z(u,v)] . ∣∂r/∂u x ∂r/∂v∣ dudv                               

∣∂r/∂u x ∂r/∂v∣ is referred to as area element 

A Vector Field’s Surface Integral 

A vector field’s surface integral is given by the below formula:

∬sF(x,y,z).dS = ∬sF(x,y,z).ndS= ∬D(u,v)F[x(u,v),y(u,v),z(u,v))] . [∂r/∂u × ∂r/∂v] dudv

Surface Integral: Applications

Surface Integrals have a wide number of applications in various fields of engineering and science. Some of its important applications when it comes to vector calculus are given below:

• You can use surface integrals to calculate a surface’s electric charge.
• It proves to be helpful to find a shell’s mass. 
• You may even use it to find the centre of mass and other values like the rotational inertia of a shell.
• Vital values in physics like the force of gravity and pressure can be easily determined with the help of surface integrals.
• You can also use it to calculate the electric field. 

1. How can a parameterised surface be described?

Solution.

You can describe a parameterised surface by the given formula:

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

2. A surface has a parameterization r (u, v) = 2 cos u, 2 sinu, v, 0

Solution. A cylinder is parameterized by the equation x² + y² = R²

Suppose that the cylinder is s then its parameterization is given as below:

r (u, v) = R cosu, R sinu, v, 0

Q: Can we find surface integrals of a closed area?

Q: Can the surface integral have a negative value?

Q: What is the difference between surface integral and line integral?

Q: Give the name of the theorem that relates line integral and surface integral?

Q: Do surface integrals change with orientation?