Example 1 : Let S be the surface and let be the outward unit normal to S. If then evaluate the integral . Solution : is a paraboloid with vertex at (0, 0, 1) as shown is Figure 7.36 Consider a closed surface S which consists of two piecewise smooth surface S and S’, where …
Example 1 : Evaluate where and S is the surface of paraboloid with axis parallel to z axis . Solution : The standard equation of paraboloid is given by where is the vertex of paraboloid Comparing given equation of paraboloid (Figure 7.31) with standard equation. The vertex is (0, 0, 9). Here also, we will …
Example 1 : Evaluate where and S is the surface of the cone above the xy plane. Solution : The general equation of cone with axis parallel to z axis and vertex at with semivertical angle is given by, The cone given by above equation is shown in figure 7.25 denotes part of cone above …
Example 1 : If . Evaluate where S is the surface of the sphere above xy plane. Solution : The surface S is sphere above xy plane as shown in figure 7.23 So, S is a open surface. But, Gauss theorem applies only to surface integral on closed surface. Had the surface S been closed, …
Example 1 : Evaluate by using Gauss divergence theorem (i) (ii) over the ellipsoid Solution : S is the ellipsoid belonging to family to level surfaces as shown in Figure 7.22. The outward drawn unit normal vector to S is given by (i) Comparing the integrals For using Gauss divergence theorem, should be continuous and …
Example 1 : Using divergence theorem, evaluate where S is the closed surface bounded by the plane z = 0, z = b and the cylinder . Solution : (This region of double integral R is given by of projection cylinder on xy plane as shown in figure 7.11) Example 2 : Evaluate over the …
Example 1 : Verify the divergence theorem for taken over the region bounded by and z = 3. Solution : Let us first calculate the volume integral The region of double integral is shown in Figure 7.4 This volume V is bounded by the surface S which is a piecewise smooth surface consisting of lower …
Suppose V is the volume bounded by a closed piecewise smooth surface S. Suppose is a vector function of position which is continuous and has continuous first partial derivatives in V. Then where is the outward drawn unit normal vector to S. In other words, the surface integral of the normal component of a vector …