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Chapter 15: Problem 1

Verify Formula (2) in Stokes鈥� Theorem by evaluating the line integral and the surface integral. Assume that the surface has an upward orientation.. $$ \begin{array}{l}{\mathbf{F}(x, y, z)=(x-y) \mathbf{i}+(y-z) \mathbf{j}+(z-x) \mathbf{k} ; \sigma \text { is the por- }} \\ {\text { tion of the plane } x+y+z=1 \text { in the first octant. }}\end{array} $$

Short Answer

Expert verified
Evaluate both integrals, they should be equal.

Step by step solution

01

Understanding Stokes' Theorem

Stokes' Theorem connects the line integral around a closed curve \( C \) to a surface integral over a surface \( \sigma \) bounded by \( C \). The theorem states: \[ \oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_{\sigma} abla \times \mathbf{F} \cdot d\mathbf{S} \]where \( abla \times \mathbf{F} \) is the curl of \( \mathbf{F} \) and \( d\mathbf{S} \) is the surface element.
02

Calculate the Line Integral

The line integral \( \oint_C \mathbf{F} \cdot d\mathbf{r} \) needs to be evaluated around the boundary of the surface. The boundary of the surface \( x+y+z=1 \) in the first octant is the triangle formed by the points (1,0,0), (0,1,0), and (0,0,1).The line integral requires parameterizing each line segment between these points and then integrating \( \mathbf{F} \) along the curve. This involves segmenting the integral along each leg of triangle (1,0,0) to (0,1,0), (0,1,0) to (0,0,1), and (0,0,1) to (1,0,0), calculating each integral separately, and then summing them up.
03

Calculate the Curl of F

The curl of \( \mathbf{F} = (x-y) \mathbf{i} + (y-z) \mathbf{j} + (z-x) \mathbf{k} \) is given by: \[ abla \times \mathbf{F} = \left( \frac{\partial (z-x)}{\partial y} - \frac{\partial (y-z)}{\partial z} \right) \mathbf{i} + \left( \frac{\partial (x-y)}{\partial z} - \frac{\partial (z-x)}{\partial x} \right) \mathbf{j} + \left( \frac{\partial (y-z)}{\partial x} - \frac{\partial (x-y)}{\partial y} \right) \mathbf{k} \]Calculate to find the components.
04

Evaluate the Surface Integral

The surface integral is \[ \iint_{\sigma} abla \times \mathbf{F} \cdot d\mathbf{S} \]where \( d\mathbf{S} \) is normal to the surface and can be found using the gradient of given plane, as \( x+y+z=1 \). For the normal pointing outward, \( \mathbf{n} = abla (x+y+z) = \langle 1,1,1 \rangle \) and the surface element is obtained as\[ d\mathbf{S} = \langle 1, 1, 1 \rangle \, dx \, dy \]Substitute \( abla \times \mathbf{F} \) and calculate the surface integral over the triangular region projected onto the xy-plane.
05

Verify Equality from Stokes' Theorem

Now, compare the line integral calculated in Step 2 to the surface integral obtained in Step 4. According to Stokes' Theorem, these two should be equal under the assumptions given in the problem. Otherwise, check calculations for common mistakes like wrong parameterizations, incorrect computations of derivatives, or misalignment in normal vector definition.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Line Integrals
A line integral extends the concept of integration to functions defined along a curve in space. This type of integral often involves integrating a vector field along a path. In simple terms, you're adding up tiny contributions of the vector field along that path.

For a vector field \( \mathbf{F} \), a line integral around a closed curve \( C \) is written as:
  • \( \oint_{C} \mathbf{F} \cdot d\mathbf{r} \)
This expression entails taking the dot product of the vector field \( \mathbf{F} \) with the small vector \( d\mathbf{r} \), which is tangent to the curve \( C \).

To compute this, you usually:
  • Parameterize the curve \( C \), representing the path by equations that give it shape.
  • Evaluate the integral by summing the scaled contributions of \( \mathbf{F} \) along \( C \).
This was done in the original exercise by considering each segment of the triangular path in the first octant.
Demystifying Surface Integrals
Surface integrals extend the idea of integration to include surfaces, summing contributions over a two-dimensional surface rather than a single curve. When the vector field \( \mathbf{F} \) interacts with a surface \( \sigma \), it creates a surface integral, capturing the flow across that surface.

The surface integral of a vector field is denoted by:
  • \( \iint_{\sigma} \mathbf{F} \cdot d\mathbf{S} \)
Here, \( d\mathbf{S} \) is an infinitesimal vector normal to the surface at each point.

To evaluate a surface integral, we:
  • Choose a parameterization of the surface that suits the structure (e.g., planes, spheres).
  • Determine the orientation of the surface to compute the correct normal vectors \( n \).
  • Calculate and integrate the dot product \( \mathbf{F} \cdot d\mathbf{S} \).
In the original exercise, this helped check the validity of Stokes' Theorem by evaluating the integral over the triangular surface in the first octant.
Exploring the Curl of a Vector Field
The curl of a vector field provides valuable insights into its rotational characteristics. Imagine the vector field as a swirling current; the curl measures how much and in which direction this current twists.

The curl is computed as:
  • \( abla \times \mathbf{F} \)
In component form, for \( \mathbf{F} = (F_1, F_2, F_3) \):
  • \( abla \times \mathbf{F} = \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z} \right) \mathbf{i} + \left( \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x} \right) \mathbf{j} + \left( \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right) \mathbf{k} \)
Finding the curl helps us apply Stokes' Theorem, converting a line integral into an equivalent surface integral.

In the exercise, we found the curl of \( \mathbf{F} \) to evaluate the essential surface integral. This step checks if the swirl of the vector field across the surface aligns with the circulation around its boundary, as Stokes' Theorem predicts.

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