Question

Verify Stokes's Theorem by evaluating $$\int_{C} \mathbf{F} \cdot \mathbf{T} \boldsymbol{d} s=\int_{C} \mathbf{F} \cdot \boldsymbol{d} \mathbf{r}$$ as a line integral and as a double integral. \(\mathbf{F}(x, y, z)=x y z \mathbf{i}+y \mathbf{j}+z \mathbf{k}\) \(S: 3 x+4 y+2 z=12,\) first octant

Short answer

By mathematical calculations for line integral and surface integral, you can verify Stokes' theorem for given vector field and surface. If the values of line integral and surface integral are equal, then Stokes's Theorem is verified.

Step-by-step solution

Express the Surface in Parametric Form

The given surface \(S: 3x + 4y + 2z = 12\) lies in the first octant, where \(x, y, z \geq 0\). It can be expressed parametrically as \(r(u, v) = (4u, 3u, 12 - 7u)\) where \(0 \leq u, v \leq 1\).

Calculate the Line Integral

The line integral of F over the curve C is defined as \(\oint_C F \cdot T ds\), where T is the unit tangent vector and ds is the differential element along the curve. However, for the given F, T is not provided, but the expression \(\oint_C F \cdot dr\) is, which is equivalent to the line integral when F is a conservative vector field. The boundary C of the surface S is the triangle in the xy-plane with vertices at (0, 0, 0), (4, 0, 0), and (0, 3, 0). Parametrizing these line segments and plugging in for \(dr\) and F, the line integral can be solved.

Calculate the Double Integral

Stokes' Theorem relates the line integral to a double integral over the surface S of the curl of F. The curl of F is calculated as \(curl(F) = \nabla \times F = (1, -2yz, 2xz - y)\). The differential surface element \(dS\) in terms of the parameters u and v is ||cross product of partial derivatives of r with respect to u and v|| = \(7du dv\). Therefore, the double integral becomes \(\int_S curl(F) \cdot dS\), which can be carried out using the calculated curl vector and the parametrization of the surface.