Integral Theorems Civil Engineering (CE) Notes | EduRev

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Civil Engineering (CE) : Integral Theorems Civil Engineering (CE) Notes | EduRev

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Divergence Theorem 
Suppose u(x) is a vector field defined on some convex three dimensional region, R in physical space with volume v and on a closed surface a bounding this volume and u(x) is continuous in R and continuously differentiable in the interior of R. Then,

Integral Theorems Civil Engineering (CE) Notes | EduRev

where, n is the outward unit normal field acting along the surface a, dv and da are infinitesimal volume and surface element, respectively. Proof of this theorem is beyond the scope of these notes.
Using (2.262) the following identities can be established:

Integral Theorems Civil Engineering (CE) Notes | EduRev

where Φ, u, A are continuously differentiable scalar, vector and tensor fields defined in v and continuous in v and n is the outward unit normal field acting along the surface a.


Stokes theorem 
Stokes theorem relates a surface integral, which is valid over any open surface a, to a line integral around the bounding single simple closed curve c in three dimensional space. Let dx denote the tangent vector to the curve, c and


Integral Theorems Civil Engineering (CE) Notes | EduRev

denote the outward unit vector field n normal to s. The curve c has positive orientation relative to n in the sense shown in figure 2.3. The indicated circuit with the adjacent points 1, 2, 3 (1, 2 on curve c and 3 an interior point of the surface s) induced by the orientation of c is related to the direction of n (i.e., a unit vector normal to s at point 3) by right-hand screw rule. For a continuously differentiable vector field u defined on some region containing a, we have

Integral Theorems Civil Engineering (CE) Notes | EduRev

Let Φ and u be continuously differentiable scalar and vector fields defined on a and c and n is the outward drawn normal to a. Then, the following can be show :Integral Theorems Civil Engineering (CE) Notes | EduRev


Green’s theorem 
Finally, we record Greens’s theorem. We do this first for simply connected domain and then for multiply connected domain. If any curve within the domain can be shrunk to a point in the domain without leaving the domain, then such a domain is said to be simply connected. A domain that is not simply connected is said to be multiply connected.

Applying Stoke’s theorem, (2.268) to a planar simply connected domain with the vector field given as u = f(x, y)e1 + g(x, y)e2,

Integral Theorems Civil Engineering (CE) Notes | EduRev

The above equation (2.271) when f(x, y) = 0 reduces to

Integral Theorems Civil Engineering (CE) Notes | EduRev

where θ is the angle made by the tangent to the curve c with e1 and nis the component of the unit normal n along e1 direction, refer to figure 2.4. Then, when g(x, y) = 0 equation (2.271) yields,
Integral Theorems Civil Engineering (CE) Notes | EduRev

where ny is the component of the unit normal along the e2 direction. Similarly, applying Stoke’s theorem (2.268) to a vector field, Integral Theorems Civil Engineering (CE) Notes | EduRev  Integral Theorems Civil Engineering (CE) Notes | EduRev   defined over a planar simply connected domain,

Integral Theorems Civil Engineering (CE) Notes | EduRev

The above equation (2.274) when Integral Theorems Civil Engineering (CE) Notes | EduRev reduces to

Integral Theorems Civil Engineering (CE) Notes | EduRev

where ny is the component of the normal to the curve along the e2 direction and when Integral Theorems Civil Engineering (CE) Notes | EduRev yields

Integral Theorems Civil Engineering (CE) Notes | EduRev

where nz is the component of the normal to the curve along the e3 direction. 

Integral Theorems Civil Engineering (CE) Notes | EduRev
Figure 2.4: Relation between tangent and normal vectors to a curve in xy plane

Integral Theorems Civil Engineering (CE) Notes | EduRev
Figure 2.5: Illustration of a multiply connected region 

In a simply connected domain, the region of interest has a single simple closed curve. In a multiply connected domain, the region of interest has several simple closed curves, like the one shown in figure 2.5. If the line integrals are computed by traversing in the direction shown in figure 2.5,

Integral Theorems Civil Engineering (CE) Notes | EduRev

where the functions f and g are continuously differentiable in the domain, C0 is the outer boundary of the domain, Ci ’s are the boundary of the voids in the interior of the domain and we have assumed that there are N such voids in the domain. We emphasize again our assumption that we transverse the interior voids in the same sense as the outer boundary.
The above equation (2.277) when f(x, y) = 0 reduces to

Integral Theorems Civil Engineering (CE) Notes | EduRev

where nix is the component of the unit normal n along e1 direction of the i th curve. Then, when g(x, y) = 0 equation (2.277) yields,

Integral Theorems Civil Engineering (CE) Notes | EduRev

where niy is the component of the unit normal along the e2 direction of the ith curve.

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