Line integral of a vector field

• realcomfy
In summary, the conversation was about calculating a line integral of a vector field using cartesian coordinates. The speaker encountered difficulties with their calculations, specifically with an incorrect substitution in their integral. The correct substitution is x = sqrt(R^2 - y^2), which gives the correct result of Pi*R^4/4.
realcomfy
Hello, I am attempting to calculate the line integral of the vector field Line integral of a vector field $$\overline{A}= x^{2} \hat{i} + x y^{2} \hat{j}$$ around a circle of radius R ($$x^{2} + y^{2} = R^{2}$$) using cartesian coordinates.

The appropriate differential line element in cartesian coordinates is $$dx \hat{i} + dy \hat{j}$$ and after taking the dot product between the two I am left with

$$\int x^{2} dx + x y^{2} dy$$

For the first term, I integrated x from -R to R (its max and min on the circle) while for the second term I set $$x = [tex]\sqrt{R^{2} - y^{2}}$$, leaving me with $$\int y^{2}* \sqrt{R^{2} - y^{2}} dy$$ and integrated over the same region. The first term was easy, and after some substitutions I was able to get the second term as well. However, I was left with 2*R^3/3 + Pi*R^4/8 and the answer was supposed to be simply Pi*R^4/4

So the second term is off by a factor of 1/2 and the first shouldn't even be there. Also, the second was only obtained by plugging in the values of an arctan term into Wolfram alpha, a method I am assuming I shouldn't have to use. If anyone can spot where I am making the error or if I am performing an operation that isn't allowed it would be greatly appreciated. Thanks for your time.

It looks like you are making an incorrect substitution in your integral. The correct substitution is x = sqrt(R^2 - y^2). This yields the following integral: \int -y^2\sqrt{R^2-y^2}dyIntegrating this will give you the correct result of Pi*R^4/4.

1. What is a line integral of a vector field?

A line integral of a vector field is a mathematical concept used to calculate the total effect of a vector field along a specific curve or path. It involves taking the dot product of the vector field and a small segment of the curve, and then summing these products along the entire curve.

2. What is the difference between a line integral and a regular integral?

A regular integral calculates the area under a curve, while a line integral calculates the total effect of a vector field along a specific curve. In other words, a regular integral is a 2-dimensional concept, while a line integral is a 3-dimensional concept.

3. How is a line integral of a vector field calculated?

To calculate a line integral of a vector field, you first need to parameterize the curve along which the integral is being calculated. This means expressing the curve as a function of one variable. Then, you take the dot product of the vector field and the derivative of the curve with respect to that variable, and integrate this product over the limits of the curve.

4. What are some real-world applications of line integrals of vector fields?

Line integrals of vector fields have many practical applications in fields such as physics, engineering, and economics. They are used to calculate work done by a force, fluid flow, electrical potential, and many other physical quantities that can be represented by vector fields.

5. Are there any alternative methods for calculating line integrals of vector fields?

Yes, there are alternative methods for calculating line integrals of vector fields, such as the Green's theorem, Stokes' theorem, and the Divergence theorem. These theorems provide a more efficient way of calculating line integrals by converting them into surface or volume integrals, which can be easier to evaluate in certain cases.

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