What is the significance of the negative direction in Green's Theorem?

In summary, the speaker is having trouble with displaying mathematical formulas and is seeking help on understanding the green theorem. They are confused about the direction of C and where the '2t dt' term comes from in integration.
  • #1
cersepn
1
0
After much trouble with using the in-built engine to display all the mathematical formula, i thought scanning the question was the only way i could get my point across

For (1) is C also defined by the parametric representation of [tex]\widetilde{}C[/tex] which is why the direction of C is negative?
Then for (3), i don't get where the '2t dt' term comes from

Help would be much appreciated.. I've been searching online for some good green theorem lessons but I'm still kinda confused after reading them all.. thanks

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  • #2
again!For (1), yes, C is also defined by the parametric representation of \widetilde{}C. The direction of C is negative because the parametric representation is given as (x, y) = (t, -t). This means that, as t increases, the x-coordinate increases in the positive direction, while the y-coordinate decreases in the negative direction. For (3), the '2t dt' term comes from the integration of the function f(t) = 2t. The integral of f(t) can be written as \int f(t) dt = \int 2t dt = 2t^2/2 + c, where c is a constant. To simplify this, we can drop the constant term, leaving us with 2t dt.
 

FAQ: What is the significance of the negative direction in Green's Theorem?

What is Green's Theorem?

Green's Theorem is a fundamental theorem in vector calculus that relates the line integral of a two-dimensional vector field over a closed curve to the double integral of the divergence of the vector field over the enclosed region.

Why is Green's Theorem important?

Green's Theorem is important because it provides a powerful tool for solving line and surface integrals, which are essential in many areas of science and engineering, including physics, fluid mechanics, and electromagnetism.

How is Green's Theorem used in real-world applications?

Green's Theorem has many real-world applications, such as calculating the work done by a force on a particle moving along a closed path, finding the flow of a fluid through a given region, and computing the electric flux through a closed surface.

What are the prerequisites for understanding Green's Theorem?

To understand Green's Theorem, one should have a basic understanding of vector calculus, including vector fields, line integrals, and double integrals. Knowledge of calculus and multivariable calculus is also necessary.

Are there any limitations to using Green's Theorem?

Green's Theorem can only be applied to two-dimensional vector fields and closed curves. It also assumes that the vector field and the enclosed region are well-behaved and continuous. In some cases, other methods, such as Stokes' Theorem, may be needed to solve a problem that cannot be solved using Green's Theorem.

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