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Divergence Theorem is a theorem that is used to compare the surface integral with the volume integral. It helps to determine the flux of a vector field via a closed area to the volume encompassed in the divergence of the field. It is also known as Gauss's Divergence Theorem in vector calculus.
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Key Takeaways: Gauss divergence theorem, surface integral, volume integral, coordinate plane, coordinates, abscissa, ordinate, vector, volume
Divergence Theorem Statement
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The Divergence Theorem states that the surface integral of the normal component of a vector point function “F” over a closed surface “S” is equivalent to the volume integral of the divergence of taken over the volume “V” encircled by the surface S. Symbolically, the divergence theorem is represented by the following equation:
Proof of Divergence Theorem
Let us assume a closed surface represented by S which encircles a volume represented by V. Any line drawn parallel to the coordinate axis intersects S at nearly two points.
Let S1 and S2 be the surfaces at the top and bottom of S, denoted by z=f(x,y) and z=\(\theta\)(x,y), respectively.
So, for the upper surface S2,
So the normal vector n1 to S1 makes a obtuse angle,
Hence, the above expression can be written as,
Similarly, projecting the surface S on coordinate plane,
Now, by adding the above all three equations,
Hence, proved.
Applications of Divergence Theorem
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Divergence Theorem finds its applications in the field of calculus including -
- In vector fields where the inverse-square law is followed including electrostatic field, gravity, and quantum physics.
- It is used in the field of calculus to obtain the flux of the vector field through a closed area to the volume encircled in the divergence field.
Things to Remember
- The Divergence Theorem compares the surface integral with the volume integral.
- The Divergence Theorem states that the outward flux via a closed surface is similar to the integral volume in the surface of the divergence area.
- The Divergence Theorem gives the flux by the closed surface of a vector field to the divergence in the circled volume of the area.
- Mathematically, the divergence theorem is denoted by the following equation:
Divergence Theorem
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Sample Questions
Ques: Use Divergence theorem to find S is the surface of the unit sphere with its center at the origin and the positive direction is the outward normal. (5 marks)
Ans: By applying Gauss's theorem to find the flux of the given vector field across the close surface (surface integral).
F (x, y, z) = (x - y, y2 + z2, y - x2) - Vector field
According to the Divergence theorem,
Applying spherical coordinates we can define the solid D (unit sphere with its centre at the origin)
x = ρsin\(\emptyset\)cosθ
y = ρsin\(\emptyset\)sinθ
z = ρcos\(\emptyset\)
V = {(ρ,θ,\(\emptyset\))|0 ≤ ρ ≤1 ,0 ≤ θ ≤ 2π, 0 ≤ \(\emptyset\) ≤π} - (Region in spherical coordinates)
To obtain the triple integral,
Jsc = ρ2sin\(\emptyset\) (Jacobian of the transformation)
Hence, the result of the surface integral by applying Gauss's theorem is 4/3 \(\pi\)
Ques: Evaluate the surface integral \(\iint\) (3x i + 2y j). dS where S is the surface given by x2 + y2 + z2 = 9. (2 marks)
Ans: Here, the divergence theorem is given by
\(\iint\) F.dS = \(\iiint\) Div (F). dV
Div (3x i + 2y j)
3+2 = 5
Now, the volume integral will be
\(\iiint\)5.dV, where dV is the volume of the sphere 4 \(\prod\) r3 /3 and r = 3 units.
Hence, we obtain 180\(\pi\).
Ques: Evaluate the divergence theorem for a function given by F = 4x i + 7y j + z k, if the surface considered is a cone of radius ½ \(\Pi\)m and height 4 \(\Pi\)2 m. (3 marks)
Ans: Div (F) = 4+7+1 = 12.
By using the divergence theorem,
\(\iiint\) (12).dV, where dV is the volume of the cone \(\Pi\)r3h/3,
where r = ½ \(\pi\) m and
h = 4 \(\pi\)2 m.
By applying the substitution method in the radius and height in the triple integral,
we get 2 units.
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