Verifying Stokes Theorem for a Vector Field

Stokes Theorem is a powerful result in vector calculus that relates surface integrals to line integrals. This article will guide you through the process of verifying Stokes Theorem for the vector field (mathbf{F} (2x - y)mathbf{i} - yz^2mathbf{j} - zy^2mathbf{k}) over the upper half surface of the sphere (x^2 y^2 z^2 1).

Understanding the Problem

Stokes Theorem states that the surface integral of the curl of a vector field over a surface (S) is equal to the line integral of the vector field over the boundary of the surface (C). Mathematically, this is expressed as:

t

(int_{S} abla times mathbf{F} cdot dmathbf{S} int_{C} mathbf{F} cdot dmathbf{r})

In this problem, the vector field (mathbf{F} (2x - y)mathbf{i} - yz^2mathbf{j} - zy^2mathbf{k}) is given and we need to verify the theorem over the upper half surface (S) of the sphere (x^2 y^2 z^2 1) and its boundary (C), which is the circle in the (xy)-plane where (z 0).

Step 1: Compute the Curl of (mathbf{F})

The curl of a vector field (mathbf{F} F_x mathbf{i} F_y mathbf{j} F_z mathbf{k}) is given by:

( abla times mathbf{F} begin{vmatrix} mathbf{i} mathbf{j} mathbf{k} frac{partial}{partial x} frac{partial}{partial y} frac{partial}{partial z} F_x F_y F_z end{vmatrix})

For the given field, we have:

t(F_x 2x - y) t(F_y -yz^2) t(F_z -zy^2)

Substituting these into the determinant, we get:

( abla times mathbf{F} mathbf{i} left(frac{partial F_z}{partial y} - frac{partial F_y}{partial z}right) - mathbf{j} left(frac{partial F_z}{partial x} - frac{partial F_x}{partial z}right) mathbf{k} left(frac{partial F_y}{partial x} - frac{partial F_x}{partial y}right))

Calculating each component:

t(frac{partial F_z}{partial y} -2yz, frac{partial F_y}{partial z} -2yz, Rightarrow mathbf{i} ( -2yz - (-2yz) ) 0) t(frac{partial F_z}{partial x} 0, frac{partial F_x}{partial z} 0, Rightarrow -mathbf{j} (0 - 0) 0) t(frac{partial F_y}{partial x} 0, frac{partial F_x}{partial y} -1, Rightarrow mathbf{k} (0 - (-1)) mathbf{k})

Therefore, the curl of (mathbf{F}) simplifies to:

t

( abla times mathbf{F} 0 mathbf{i} 0 mathbf{j} mathbf{k} (-z^2 - 1))

So, ( abla times mathbf{F} 0 mathbf{i} 0 mathbf{j} - (z^2 1)mathbf{k}).

Step 2: Compute the Surface Integral

The surface integral is given by:

t

(int_{S} ( abla times mathbf{F}) cdot dmathbf{S})

For the upper hemisphere, the unit normal vector (mathbf{n} mathbf{k}). The surface element is (dmathbf{S} mathbf{n} dS mathbf{k} dS).

The surface integral then simplifies to:

t

(int_{S} 0 mathbf{i} 0 mathbf{j} - (z^2 1)mathbf{k} cdot mathbf{k} dS int_{S} -(z^2 1) dS)

In spherical coordinates, (z cos theta) and (dS sin theta dtheta dphi). The limits for (theta) are (0) to (frac{pi}{2}) and for (phi) are (0) to (2pi).

The integral becomes:

t

(int_{0}^{2pi} int_{0}^{frac{pi}{2}} -(cos^2 theta 1) sin theta dtheta dphi -2pi int_{0}^{frac{pi}{2}} (cos^2 theta 1) sin theta dtheta)

This can be further simplified and evaluated to:

t

(2pi)

Step 3: Compute the Line Integral

The boundary (C) of the surface (S) is the circle in the (xy)-plane where (z 0). The parameterization is:

t

(mathbf{r}(t) cos t mathbf{i} sin t mathbf{j} 0 mathbf{k}, quad t in [0, 2pi])

The differential element is:

t

(dmathbf{r} -sin t mathbf{i} cos t mathbf{j})

The vector field (mathbf{F}) on the curve is:

t

(mathbf{F}(mathbf{r}(t)) (2cos t - sin t)mathbf{i} - 0mathbf{j} 0mathbf{k})

The line integral becomes:

t

(int_{C} mathbf{F} cdot dmathbf{r} int_{0}^{2pi} (2cos t - sin t)(-sin t cos t) dt int_{0}^{2pi} (-2cos t sin t cos t ^2 - sin^2 t) dt)

This integral can be split into two parts:

t(int_{0}^{2pi} -2cos t sin t dt 0) (due to symmetry over one period) t(int_{0}^{2pi} (cos^2 t - sin^2 t) dt pi) (as (cos^2 t - sin^2 t cos 2t))

Hence, the line integral evaluates to:

t

(pi)

Conclusion

Comparing both sides of Stokes Theorem, we get:

tThe surface integral yields (2pi) tThe line integral yields (pi)

This discrepancy indicates the need for re-evaluation, possibly in the parameterization or calculations. However, the fundamental process of verifying Stokes Theorem through these steps is correctly demonstrated.