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Chapter 10: Problem 13

Calculate this line integral by Stokes's theorem, clockwise as seen by a person standing at the origin, for the following \(F\) and \(C\) Assume the Cartesian coordinates to be right handed. (Show the details.) \(\mathbf{F}=\left[\begin{array}{lll}y^{2} & x^{2} & -x+z\end{array}\right],\) around the triangle with vertices (0,0,1),(1,0,1),(1,1,1)

Short Answer

Expert verified
The line integral is \(-\frac{1}{3}\).

Step by step solution

01

Determine Surface and Boundary

The triangle described by the vertices (0,0,1), (1,0,1), and (1,1,1) lies in the plane defined by z = 1. The boundary curve C is the perimeter of this triangle.
02

Parametrize the Boundary Curve

We have three edges along the triangle: from (0,0,1) to (1,0,1), from (1,0,1) to (1,1,1), and from (1,1,1) to (0,0,1). The parametrizations are: \[ \begin{align*}\mathbf{C}_1 &: (1,0,1)t \, \text{for} \, t\in[0,1], \\mathbf{C}_2 &: (1,t,1) \, \text{for} \, t\in[0,1], \\mathbf{C}_3 &: (1-t,1,1) \, \text{for} \, t\in[0,1].\end{align*}\]
03

Find the Curl of F

The curl of a vector field \(\mathbf{F} = (F_1, F_2, F_3)\) is given by \(abla \times \mathbf{F}\). For \(\mathbf{F} = (y^2, x^2, -x+z)\),\[abla \times \mathbf{F} = \left(\frac{\partial (-x+z)}{\partial y} - \frac{\partial (x^2)}{\partial z}, \frac{\partial (y^2)}{\partial z} - \frac{\partial (-x+z)}{\partial x}, \frac{\partial (x^2)}{\partial y} - \frac{\partial (y^2)}{\partial x}\right).\]Calculating, we find:\[abla \times \mathbf{F} = (0, 1, -2y).\]
04

Calculate the Surface Integral

Using Stokes' Theorem,\[\oint_{C} \mathbf{F} \cdot d\mathbf{r} = \int_{S} (abla \times \mathbf{F}) \cdot d\mathbf{S}.\]Here, \(S\) is the triangle in the plane \(z=1\), and \(d\mathbf{S} = \hat{\mathbf{k}}\,dA\). Since \(abla \times \mathbf{F} = (0, 1, -2y)\), we calculate the dot product: \[(abla \times \mathbf{F}) \cdot \hat{\mathbf{k}} = -2y.\]Integrate over the area of the triangle:\[\int_{0}^{1}\int_{0}^{x}(-2y)\,dy\,dx = -2 \int_{0}^{1} \left. \left( \frac{y^2}{2} \right) \right|_{0}^{x} \,dx = - \int_{0}^{1} x^2 \,dx = -\frac{1}{3} x^3 \bigg|_0^1 = -\frac{1}{3}.\]
05

Interpret the Result

The negative sign in the final result indicates the direction is opposite to our assumed positive orientation for the loop. Since we calculated based on the clockwise path as viewed from the origin, the line integral evaluated as intended.

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Key Concepts

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

Line Integral
A line integral is a fundamental concept in vector calculus. It represents the integral of a function along a specified curve. This type of integral is typically used for functions that describe physical quantities, such as force fields. In our exercise, we're using the line integral to measure the work done by the vector field \( \mathbf{F} = \left(y^2, x^2, -x+z\right) \) along the boundary of a triangle. The boundary, or curve \(C\), is parametrized to account for each segment of the triangle lying on a plane. This process ensures that each edge is carefully considered. When dealing with line integrals, remember:
  • The curve's orientation matters; it must align with the vector field's direction to correctly represent work or flow.
  • Parametrizing the curve allows for explicit representation of the path in terms of a variable, often \(t\).
  • Stokes's Theorem provides a way to calculate a line integral through a surface integral, making some computations simpler, especially in three dimensions.
Vector Calculus
Vector calculus is the branch of mathematics that deals with vector fields and their operations. Concepts like divergence, gradient, and curl are central. These tools help analyze fields that have both magnitude and direction, such as electromagnetic or gravitational fields.In the context of our exercise, vector calculus is applied to find the curl of vector field \(\mathbf{F}\). The curl is a vector operation that describes the rotation of the field. Using the notation \(abla \times \mathbf{F}\), the curl provides insight into how the field behaves locally:
  • The curl \(abla \times \mathbf{F} = (0, 1, -2y)\) was calculated using partial derivatives. This value is critical for surface integrals.
  • Understanding the curl helps determine how a tiny loop within \(\mathbf{F}\) behaves, influencing the surface integral.
  • Vector calculus emphasizes understanding field dynamics, critical for applying Stokes's Theorem efficiently.
Parametrization
Parametrization is the process of defining a curve using parameters. It transforms the complex geometric shape into a simpler mathematical form. Parametrization is particularly useful in line integrals and when performing operations like integration over paths.In the exercise, each edge of the triangular path is parametrized:
  • \( \mathbf{C}_1: (1,0,1)t \) for \( t \in [0,1] \)
  • \( \mathbf{C}_2: (1,t,1) \) for \( t \in [0,1] \)
  • \( \mathbf{C}_3: (1-t,1,1) \) for \( t \in [0,1] \)
These parametrizations efficiently describe each segment, enabling precise integration over the curve:
  • Parametrizing simplifies calculations by converting geometry into algebraic equations.
  • It provides a tangible way to "walk" along the curve, making integration feasible.
  • All ensure the directions align with the chosen orientation for integration, crucial for accurate results.

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Most popular questions from this chapter

Evaluate the integral \(\iint_{S}(\operatorname{curl} \mathbf{F}) \cdot \mathbf{n} d A\) directly for the given \(\mathbf{F}\) and \(\mathbf{S}\). $$\mathbf{F}=\left[\begin{array}{lll}4 x^{2}, & 16 x, & 0], S ; z=y(0 \leqq x \leqq 1,0 \leq y \leq 1)\end{array}\right.$$

Calculate this line integral by Stokes's theorem, clockwise as seen by a person standing at the origin, for the following \(F\) and \(C\) Assume the Cartesian coordinates to be right handed. (Show the details.) $$\begin{aligned} &\mathbf{F}=[\cos \pi y, \quad \sin \pi x, \quad 0], \text { around the rectangle with }\\\ &\text { vertices }(0,1,0),(0,0,1),(1,0,1),(1,1,0) \end{aligned}$$

Integrate \(x y e^{-x^{2}-y^{2}}\) over the triangular region with vertices (0,0),(1,1),(1,2)

Using Green's theorem, evaluate \(\int_{c} \mathbf{F}(\mathbf{r}) \cdot d \mathbf{r}\) counterclockwise around the boundary curve \(C\) of the region \(R\), where $$\begin{aligned} &\mathbf{F}=\left[x^{2}+y^{2}, x^{2}-y^{2}\right], R: 1 \leq y \leq 2-x^{2} . \text { Sketch }\\\ & R. \end{aligned}$$

and, if independent, integrate from (0,0,0) to \((a, b, c)\). $$x y z^{2} d x+\frac{1}{2} x^{2} z^{2} d y+x^{2} y z d z$$
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