Multivariable Calculus

Notes on lecture 19



1 Surface integral of vector fields
R R
Earlier, when we studied ordinary surface integrals, we computed S f (x, y, z)dS
for some scalar function f over a surface S. Now, this scalar function
f (x, y, z) is going to be replaced by the dot product of some vector func-
tions (which being a dot product, is again a scalar function).
We are going to consider a surface S and a point P (x, y, z) on it. At the
point P , we are going to consider an outward unit normal vector n̂. Sup-
pose we define a vector field F~ over the surface S (that is, at each point of
S, we are able to define a vector-valued function F~ ), then the dot product
~
R ·Rn̂ is going to be a scalar function and we are simply going to compute
F
S
F~ · n̂dS. That is, the integrand f (x, y, z) in ordinary surface integrals is
going to be replaced here by F~ · n̂. Figure 1.0.1 illustrates this.




Figure 1.0.1: Surface integral of vector fields

For computing the surface integral of the vector field F~ over a surface S, we

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,are going to proceed as follows:

We divide the surface S into patches, and on each patch Sij , we consider
the product
(F~ · n̂)(x⇤ij , yij⇤ , zij⇤ )A(Sij ),
where A(Sij ) is the area of the patch Sij and (x⇤ij , yij⇤ , zij⇤ ) is a point on Sij .
Then we take a double sum of the above expression over all i, j, and take
limits as the number of patches tend to 1. That will give us the surface
integral of the vector field F~ over S.

1.1 A surface immersed in a fluid
We are going to assume that the surface S is like an imaginary net which is
immersed in some fluid and the fluid is flowing over that imaginary surface.
Assume also that S is a closed surface. Consider a patch on the surface S.
Then we will be able to draw an imaginary column inside the surface (as
shown in Figure 1.1.1).




Figure 1.1.1: An imaginary column inside the surface S

Then the fluid is going to enter the surface through the patch (that we had
considered), pass through the column, and it is going to come out of the
column from the other side of the surface.




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, 1.2 A digression
At this point, let us recall the concept of scalar projection. Suppose we want
to consider the scalar projection of a vector ~b over another vector ~a. Let ✓
be the angle between ~a and ~b. Then if l denotes the scalar projection of ~b
over ~a, we have
l
= cos ✓.
|~b|
This implies that
l = |~b| cos ✓ = |~b||â| cos ✓ = ~b · â,
where â is the unit vector along ~a.
Now suppose we are considering some vector field F~ and we want to compute
the component of F~ along a particular direction ~a, then that component is
simply going to be given by F~ · â, where â is the unit vector along ~a.

1.3 Back to a surface immersed in a fluid
Let us once again consider the surface S immersed in a fluid. Suppose the
fluid is moving with a velocity ~v and n̂ is a unit normal vector at a point on
the patch, then the component of ~v along the direction of n̂ is given by ~v · n̂.
We know from physics that
distance = velocity ⇥ time.
Now, if we assume that t is the time taken by the fluid to cross the column,
then the length of the column will be given by (~v · n̂) t. And hence the
volume of the column is given by {(~v · n̂) t}dS, where dS denotes the area
of the patch.
We know from physics that
density = mass per unit volume.
Then the mass of the fluid inside the column for time t is given by
⇢{(~v · n̂) t}dS,
where ⇢ denotes the density of the fluid. That means mass of fluid contained
in the column per unit time is going to be
⇢{(~v · n̂) t}dS
= ⇢(~v · n̂)dS.
t
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