Determine which theorem applies to simplify the following. It is possible that more than one will apply. Find the work done by the vector field F(r, y) = yz cos(ry)i + rz cos(ry)j+ sin(zy)k along the positively oriented triangle with vertices (2,1,-6), (3,-1,1), and (0,1,0). %3D O Fundamental Theorem of Line Integrals O Green's Theorem O Stoke's Theorem O The Divergence Theorem
Arc Length
Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic path. Alternatively, if a curve on a map represents a road, we might want to know how far we need to drive to get to our destination. The distance between two points along a curve is known as arc length.
Line Integral
A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along a curve is very important in mathematics. Usually, by a line integral, we compute the area of the function along the curve. This integral is also known as curvilinear, curve, or path integral in short. If line integrals are to be calculated in the complex plane, then the term contour integral can be used as well.
Triple Integral
Examples:
![### Problem: Applying Theorems to Simplify the Calculation of Work Done by a Vector Field
**Instructions**:
Determine which theorem applies to simplify the following problem. It is possible that more than one theorem will apply.
**Question**:
Find the work done by the vector field \( \mathbf{F}(x, y) = yz \cos(xy) \mathbf{i} + xz \cos(xy) \mathbf{j} + \sin(xy) \mathbf{k} \) along the positively oriented triangle with vertices \((2,1,-6)\), \((3,-1,1)\), and \((0,1,0)\).
**Check the applicable theorems**:
- [ ] Fundamental Theorem of Line Integrals
- [ ] Green's Theorem
- [ ] Stokes' Theorem
- [ ] The Divergence Theorem
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**Interactive Elements**:
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**Note**:
Using proper theorems for vector fields and line integrals simplifies complex calculations, enhancing understanding and efficiency in solving such problems in vector calculus.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Facdec92e-028d-4f37-af60-841922a4cee5%2F84f2eaf8-7551-4450-996e-ce67bf5ca897%2Fmvihcd_processed.jpeg&w=3840&q=75)
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