Week 2 Lecture 2 Boundary Theorems.

In todays lecture we covered the remaining two boundary theorems due to Gauss and Stokes. Remember the first boundary theorem was related to the line integral of the gradient of a scalar field ( grad φ ( x, y, z) which is a vector field) and it could be seen that the line integral was independent of the path and dependent only on the value of φ at the end ( boundary) points of the path.


The second boundary theorem ( Gauss Divergence Theorem ) involves the divergence of a vector field ∇.v(x, y, z)where v is a vector field. The theorem states that the flux of the vector field
over a CLOSED SURFACE S which is the boundary of a volume V is equal to the volume integral of div v ( x, y, z) over the volume V.


Proof: First we proved that the flux of a vector field v over the surface of an infinitesimal cube sides dx, dy, dz was equal to the div v times the volume element
dτ.


Then we considered a finite CLOSED surface S bounding a finite volume V. Using the theorem that flux over a surface S is equal to the total sum of the fluxes from subdivision of S into smaller surfaces ( because as we saw adjacent surface contributions to the flux mutually cancels, leaving only flux contribution from the main surface S). The flux of the vector field v over the finite surface S would just
be the sum of the fluxes through the large number of infinitesimal cube into which the surface can be subdivided. Now the flux from each infinitesimal cube was equal to the div v times the infinitesimal volume. When this is summed over and in the limit the number of cubes become large is just the volume integral of the div V. Hence the prrof.


The last boundary theorem ( Stokes Theorem ) relates the CIRCULATION ( line integral over a CLOSED CURVE
C) of a vector field C ( x, y, z) to the surface integral of the normal component of
the Curl of the vector field over an OPEN SURFACE which has the closed curve Γ as its boundary. ( Remember that the boundary of an OPEN SURFACE is a CLOSED CURVE Γ. Also i can construct an infinite number of open surfaces which have the same closed curve Γ as their boundary.)


Proof: For the proof we first took an infinitesimal square on the (y, z) plane with area element dydz and showed that the circulation of a vector field C ( x, y,z) over this infinitesimal square area element was equal to the normal component of the Curl of the vector field C times the area element dydz ( the normal component in this case is the z component of the Curl C which is the normal component to the area element dydz when the circulation is taken in the anticlockwise direction.)


Now we may consider an arbitrary open surface S with a boundary Γ and subdivide it into a large number of infinitesimal squares. Using the theorem that the circulation over a curve Γ is equal to the sum of the circulations over the smaller curves that Γ may be subdivided into, we see that the contributions to the circulation comes only from the boundary curve, ( the contributions from adjacent squares cancel out leaving only the boundary contribution.)


However the circulation over each such infinitesimal square is equal to the normal component of Curl C times the area element. When we sum over all the area elements,
in the limit the number of such elements becoming very large the RHS is the surface integral of the normal component of the Curl C. Hence the proof.