Math Problem Statement
G3.) Consider the vector field on the domain vec F = \langleP, Q\rangle = \langle(- y)/((x - y) ^ 2), x/((x - y) ^ 2)\rangle; mathcal D =\ (x, y) in mathbb R ^ 2 |y<x\ . (i) (1 pt) Make a picture of the set D. Is D simply connected? (ii) (1 pt) Show that partial P partial y = partial Q partial x on the domain D. (iii) (1 pt) Based on the partial derivative test from part (ii) and your answer to (i), is it safe to conclude that vec F is conservative on the set D? Explain your answer. (Your explanation should reference the type of domain you have here.) (iv) (2 pts) Using the standard procedure for finding potentials, find a potential function g for vec F (v) (1 pt) Using the fundamental theorem of calculus for line integrals and part (iv), compute the integral int gamma vec F * d vec r , where gamma is a curve in the domain D starting at the point (0, - 1) and terminating at the point (1,0).
Solution
Let's go through each part step-by-step.
Part (i): Picture of the Set and Determining if is Simply Connected
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Definition of : The domain is defined as the set of points in where . This inequality represents a half-plane to the left of the line , but does not include this line.
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Simply Connected: A region in is simply connected if every closed curve within the region can be continuously contracted to a point without leaving the region. Here, (the half-plane ) is open, connected, and does not contain any holes or disjoint parts. Therefore, is simply connected.
Part (ii): Show on
To verify if the vector field is conservative, we need to check if holds throughout .
- Partial Derivatives:
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Compute : Differentiating with respect to :
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Compute : Differentiating with respect to :
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Since holds, we have verified that this condition is met in .
Part (iii): Is Conservative on ?
From part (ii), we found that holds in . Additionally, since is simply connected (from part (i)), these conditions imply that is conservative on . This is because in simply connected domains, if holds everywhere, then must be conservative.
Part (iv): Finding a Potential Function for
Since is conservative, we can find a potential function such that .
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Integrate with respect to :
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Verify with : Differentiating with respect to , we get: [ \frac{\partial g}{\partial y} = -
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Math Problem Analysis
Mathematical Concepts
Vector Fields
Conservativeness
Simply Connected Domains
Potential Functions
Line Integrals
Formulas
Partial derivative of P with respect to y: ∂P/∂y
Partial derivative of Q with respect to x: ∂Q/∂x
Line integral: ∫γ vec F · d vec r
Theorems
Fundamental Theorem of Calculus for Line Integrals
Test for Conservativeness of a Vector Field
Suitable Grade Level
University-level (Calculus III)
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