Problem 10
Question
In Exercises \(7-12,\) use the surface integral in Stokes' Theorem to calculate the circulation of the field \(\mathbf{F}\) around the curve \(C\) in the indicated direction. \begin{equation} \begin{array}{l}{\mathbf{F}=\left(y^{2}+z^{2}\right) \mathbf{i}+\left(x^{2}+z^{2}\right) \mathbf{j}+\left(x^{2}+y^{2}\right) \mathbf{k}} \\ {\text { C: The boundary of the triangle cut from the plane } x+y+z=1} \\ {\text { by the first octant, counterclockwise when viewed from above }}\end{array} \end{equation}
Step-by-Step Solution
Verified Answer
The circulation of \( \mathbf{F} \) around \( C \) is zero.
1Step 1: Understand the Problem
You are asked to calculate the circulation of a vector field \( \mathbf{F} \) around a curve \( C \), using Stokes' Theorem. The curve \( C \) is the boundary of a triangle in the plane \( x+y+z=1 \) in the first octant.
2Step 2: Stokes' Theorem
Stokes' Theorem states that the circulation of \( \mathbf{F} \) around the curve \( C \) is equal to the surface integral of the curl of \( \mathbf{F} \) over the surface \( S \) bounded by \( C \): \[ \oint_{C} \mathbf{F} \cdot d\mathbf{r} = \iint_{S} abla \times \mathbf{F} \cdot d\mathbf{S} \] where \( d\mathbf{S} \) is the differential surface vector.
3Step 3: Calculate the Curl of \( \mathbf{F} \)
The vector field is \( \mathbf{F} = \left(y^2 + z^2 \right) \mathbf{i} + \left(x^2 + z^2 \right) \mathbf{j} + \left(x^2 + y^2 \right) \mathbf{k} \). Calculate \( abla \times \mathbf{F} \) using the determinant: \[ abla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ y^2 + z^2 & x^2 + z^2 & x^2 + y^2 \end{vmatrix} \].
4Step 4: Evaluate the Determinant
Expanding the determinant: \[ abla \times \mathbf{F} = \mathbf{i}\left( \frac{\partial}{\partial y}(x^2 + y^2) - \frac{\partial}{\partial z}(x^2 + z^2) \right) - \mathbf{j}\left( \frac{\partial}{\partial x}(x^2 + y^2) - \frac{\partial}{\partial z}(y^2 + z^2) \right) + \mathbf{k}\left( \frac{\partial}{\partial x}(x^2 + z^2) - \frac{\partial}{\partial y}(y^2 + z^2) \right) \].
5Step 5: Simplify the Curl
Calculate the derivatives: \( \frac{\partial}{\partial y}(x^2 + y^2) = 2y \), \( \frac{\partial}{\partial z}(x^2 + z^2) = 2z \), \( \frac{\partial}{\partial x}(x^2 + y^2) = 2x \), \( \frac{\partial}{\partial z}(y^2 + z^2) = 2z \), \( \frac{\partial}{\partial x}(x^2 + z^2) = 2x \), \( \frac{\partial}{\partial y}(y^2 + z^2) = 2y \). So, \( abla \times \mathbf{F} = 0 \).
6Step 6: Setup the Integral
Since \( abla \times \mathbf{F} = 0 \), the surface integral becomes \( \iint_{S} (0) \cdot d\mathbf{S} = 0 \), implying no circulation around \( C \).
7Step 7: Conclusion
Using Stokes' Theorem, we've shown the circulation of \( \mathbf{F} \) around \( C \) is zero because the curl is zero everywhere over the surface.
Key Concepts
Surface IntegralVector FieldCurlCirculation
Surface Integral
A surface integral, in simple terms, extends the concept of an integral to functions over a surface in three-dimensional space. Unlike regular integrals which are done on a line (one dimension) or over an area (two dimensions), a surface integral is performed over a curved surface (three-dimensional). Surface integrals are an essential part of multivariable calculus because they allow for the measuring of complex physical quantities such as flux through a surface.
In the context of Stokes' Theorem, a surface integral is used to calculate the circulation of a vector field around a curve. The theorem essentially translates the difficult task of directly finding this circulation into a more manageable task of evaluating a surface integral. This is beneficial because sometimes evaluating things over a surface is simpler than doing so around a complex curve. The formula involved is:
\[ \iint_{S} abla \times \mathbf{F} \cdot d\mathbf{S} \]
In this exercise, the surface integral helps us find the circulation of the vector field \( \mathbf{F} \) around the curve \( C \), simplifying the process using the curl of \( \mathbf{F} \).
In the context of Stokes' Theorem, a surface integral is used to calculate the circulation of a vector field around a curve. The theorem essentially translates the difficult task of directly finding this circulation into a more manageable task of evaluating a surface integral. This is beneficial because sometimes evaluating things over a surface is simpler than doing so around a complex curve. The formula involved is:
\[ \iint_{S} abla \times \mathbf{F} \cdot d\mathbf{S} \]
In this exercise, the surface integral helps us find the circulation of the vector field \( \mathbf{F} \) around the curve \( C \), simplifying the process using the curl of \( \mathbf{F} \).
Vector Field
A vector field is essentially a function that assigns a vector to each point in space. Imagine having arrows in space that represent forces acting on points or flows going through regions. These arrows help visualize how a field behaves in space usually denoted by \( \mathbf{F} \).
In this exercise, the vector field \( \mathbf{F} \) is represented by:
Understanding vector fields is crucial for explaining electromagnetic fields, fluid flows, and other complex systems in physics and engineering. These fields can often guide physical processes observed in wave patterns, fluid dynamics, or electric fields, among other applications.
In this exercise, the vector field \( \mathbf{F} \) is represented by:
- \( \mathbf{F} = (y^2 + z^2) \mathbf{i} + (x^2 + z^2) \mathbf{j} + (x^2 + y^2) \mathbf{k} \)
Understanding vector fields is crucial for explaining electromagnetic fields, fluid flows, and other complex systems in physics and engineering. These fields can often guide physical processes observed in wave patterns, fluid dynamics, or electric fields, among other applications.
Curl
The curl of a vector field provides a measure of its rotational tendency in space. You might picture it as the way a field "twists" or "rotates" around a point in space. The operation itself is performed using mathematical operations like divergence and gradient in vector calculus.
The curl is crucial in determining if and how a vector field circulates around a path. In Stokes' Theorem, it plays a vital role as it relates the surface integral to the line integral, connecting the dots between the field's circulation around a loop and its rotations over a surface.
For the vector field \( \mathbf{F} \), the curl is calculated using:
\[ abla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ y^2 + z^2 & x^2 + z^2 & x^2 + y^2 \end{vmatrix} \]
In this particular exercise, calculating the curl showed the result as zero, meaning there is no net rotation or twist in the vector field over the surface considered.
The curl is crucial in determining if and how a vector field circulates around a path. In Stokes' Theorem, it plays a vital role as it relates the surface integral to the line integral, connecting the dots between the field's circulation around a loop and its rotations over a surface.
For the vector field \( \mathbf{F} \), the curl is calculated using:
\[ abla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ y^2 + z^2 & x^2 + z^2 & x^2 + y^2 \end{vmatrix} \]
In this particular exercise, calculating the curl showed the result as zero, meaning there is no net rotation or twist in the vector field over the surface considered.
Circulation
Circulation in a vector field refers to the total 'amount' the field lines wrap around a closed loop or curve. Think of how water might flow and circulate around an obstacle like a rock in a stream. The concept can be a bit abstract, but it's visualized by picturing the sum of the vector field's impact as it wraps around the curve.
Stokes' Theorem relates circulation with the surface underneath the curve. Rather than focusing only on the curve \( C \), Stokes' Theorem allows us to use the surface enclosed by \( C \) to find the circulation.
The exercise shows that by using Stokes' Theorem, the circulation of the vector field \( \mathbf{F} \) is zero because the curl across the surface is zero, indicating no actual circulation around the curve. This is due to the integral of a zero curl over the defined surface essentially being a zero sum, mathematically expressing no net rotation or circulation.
Stokes' Theorem relates circulation with the surface underneath the curve. Rather than focusing only on the curve \( C \), Stokes' Theorem allows us to use the surface enclosed by \( C \) to find the circulation.
The exercise shows that by using Stokes' Theorem, the circulation of the vector field \( \mathbf{F} \) is zero because the curl across the surface is zero, indicating no actual circulation around the curve. This is due to the integral of a zero curl over the defined surface essentially being a zero sum, mathematically expressing no net rotation or circulation.
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