Greens Theorem for negatively orientated curve

In summary, Greens Theorem is a fundamental theorem in vector calculus that relates a line integral around a closed curve to a double integral over the region enclosed by the curve. Negative orientation is important in this theorem as it allows for a more comprehensive understanding of the relationship between the two integrals. It affects the calculation by changing the sign of the line integral and can be used for any type of simple closed curve. In real-world applications, Greens Theorem can be used in fields such as physics, engineering, and economics to calculate work, fluid flow, and circulation of money.
  • #1
thrillhouse86
80
0
Hey All,

in vector calculus we learned that greens theorem can be used to solve path integrals which have positive orientation. Can you use greens theorem if you have negative orientation by 'pretending' your path has positive orientated and then just negating your answer ?

Regards,
THrillhouse
 
Mathematics news on Phys.org
  • #2
yes, the only change "negative orientation" instead of "positive orientation" makes on the integral is a change in sign. If you allow for that, there is no problem.
 

Related to Greens Theorem for negatively orientated curve

1. What is Greens Theorem for a negatively orientated curve?

Greens Theorem is a fundamental theorem in vector calculus that relates a line integral around a closed curve to a double integral over the region enclosed by the curve. When the curve is oriented counterclockwise, the theorem is stated as a positive value. For a negatively orientated curve, the theorem is stated as a negative value.

2. Why is it important to consider negative orientation in Greens Theorem?

Negative orientation is important in Greens Theorem because it allows for a more comprehensive understanding of the relationship between a line integral and a double integral. By considering both positive and negative orientations, we can accurately calculate the values of both integrals and gain a deeper understanding of the underlying principles.

3. How does negative orientation affect the calculation of Greens Theorem?

Negative orientation affects the calculation of Greens Theorem by changing the sign of the line integral. This means that the value of the line integral will be subtracted instead of added to the double integral, resulting in a negative value. It is important to take this into account when using Greens Theorem for a negatively orientated curve.

4. Can Greens Theorem be used for any type of curve?

Greens Theorem can be used for any type of simple closed curve, regardless of its orientation. This includes curves that are not smooth or differentiable, as long as they are closed and do not intersect themselves. However, the theorem may not be applicable for curves that are not simple closed curves.

5. What are some real-world applications of Greens Theorem for a negatively orientated curve?

Greens Theorem has many practical applications in fields such as physics, engineering, and economics. For example, it can be used to calculate the work done by a conservative force on a particle moving along a closed path, or to determine the net flow of a fluid through a closed surface. In economics, Greens Theorem can be used to calculate the circulation of money in a closed economic system.

Similar threads

B Oriented Surfaces & Surface Area: Investigating the Impact
    • General Math
Replies
5
Views
1K
Calculate this Integral around the Circular Path using Green's Theorem
    • Calculus and Beyond Homework Help
Replies
3
Views
457
B Is everything in math either an axiom or a theorem?
    • General Math
3
Replies
72
Views
5K
B Extending the Fundamental Theorem of Arithmetic to the rationals
    • General Math
2
Replies
35
Views
3K
B How to interpret Pascal's Triangle for negative numbers?
    • General Math
Replies
3
Views
346
Problem with the positive orientation of a surface (Stokes' theorem)
    • Calculus and Beyond Homework Help
Replies
14
Views
1K
What is the one-dimensional counterpart to the Green-Gauss theorem?
    • Calculus and Beyond Homework Help
Replies
28
Views
1K
What is the Generalized Stokes' Theorem and its Applications?
    • General Math
Replies
1
Views
11K
Green's theorem with a scalar function
    • Calculus and Beyond Homework Help
Replies
2
Views
1K
B Before understanding theorems in elementary Euclidean plane geometry
    • General Math
Replies
12
Views
1K

Hot Threads

Recent Insights

Back
Top