Prove Integral_C y dx + x dy Depends on Endpoints of Curve C - Maya

In summary, the conversation discusses the meaning of a statement involving the integral of a function with respect to variables x and y, which is dependent on the endpoints of a curve C. The integral is affected by the endpoints because they determine the limits of integration, and different endpoints result in different values for the integral. This concept is important in mathematics, particularly in the field of calculus, as it helps solve problems related to area and volume. Techniques such as the Fundamental Theorem of Calculus and substitution can be used to prove the dependence of the integral on the endpoints.
  • #1
mayaitagaki
8
0
I am soooo poor at this kind of proof problem...:cry:
Please help me out with this!

Show that

Integral_C y dx + x dy depends only on the endpoints of the arbitrary curve C.

(Hint: find a potential function f of the vector field <y, x> and use that to integrate
along a parametrization r(t) = <x(t), y(t)> of C with endpoints at t = a and t = b.)

I also attached a jpg file!

Thank you,
Maya:redface:
 

Attachments

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  • #2


have you tried the hint? finding a scalar potential should be reasonably staightforward
 

1. What is the meaning of "Prove Integral_C y dx + x dy Depends on Endpoints of Curve C - Maya"?

This statement is referring to a mathematical expression involving the integral of a function, denoted by Integral_C, with respect to the variables x and y. It is also stating that this integral is dependent on the endpoints of the curve C, which is likely a path or line in a two-dimensional space. The mention of "Maya" could refer to a person's name or a specific mathematical concept.

2. How is the integral of a function affected by the endpoints of a curve?

The integral of a function is affected by the endpoints of a curve because the value of the integral can change depending on where the curve begins and ends. This is because the endpoints determine the limits of integration, which are used to calculate the area under the curve. Different endpoints will result in different limits of integration, and therefore, a different value for the integral.

3. Can you provide an example of a curve where the integral is dependent on the endpoints?

Yes, let's consider the curve C defined by the equation y = x^2 from x = 0 to x = 1. The integral of the function y = x^2 with respect to x, evaluated from x = 0 to x = 1, is equal to 1/3. However, if we change the endpoints to x = 0 to x = 2, the integral will have a different value of 8/3. Therefore, the integral is dependent on the endpoints of the curve C.

4. How does the concept of a dependent integral tie into the field of mathematics?

The concept of a dependent integral is an important one in mathematics, particularly in the field of calculus. This is because it helps us understand how the value of an integral can change depending on the boundaries of the function. It also allows us to analyze and solve problems related to area and volume in various mathematical and scientific contexts.

5. Is there a way to prove that the integral is dependent on the endpoints?

Yes, there are various mathematical techniques and theorems that can be used to prove that the integral is dependent on the endpoints. For example, the Fundamental Theorem of Calculus states that the value of an integral can be found by evaluating the antiderivative of the function at the endpoints. This shows that the value of the integral is directly tied to the endpoints of the curve. Other techniques, such as substitution and integration by parts, can also be used to demonstrate the dependence of the integral on the endpoints.

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