Line integral of a vector field

• realcomfy
In summary, the conversation discusses the process of converting a vector field from Cartesian coordinates to cylindrical coordinates in order to calculate a line integral around a circle of radius R. It is mentioned that the transformation matrix from cylindrical to Cartesian is orthogonal, allowing for a simple conversion by dividing by the matrix. The final step is to write the Cartesian vector in terms of the cylindrical coordinates.
realcomfy
I am attempting to calculate the line integral of the 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 cylindrical coordinates.

It is simple enough to convert the x and y components to their cylindrical counterparts, but I am unsure what to do about the unit vectors. Since the line integral of a vector field contains a dot product between the vector field and the differential element, this requires the two to have the same unit vectors, right?

I found the conversion matrix for cylindrical to cartesian (http://en.wikipedia.org/wiki/Vector_fields_in_cylindrical_and_spherical_coordinates) but I'm not totally sure how to convert the other way around (from cartesian to cylindrical). Wikipedia says that the transform matrix is orthogonal, so does this mean I can simply divide it to the other side and end up with the transpose? Then write my cartesian vectors in terms of the cylindrical?

Yes, you are correct. The transformation matrix is orthogonal, which means that its inverse is the transpose. So, you can divide by the transformation matrix to obtain the transformation from Cartesian to cylindrical. Once you have done that, you can write your Cartesian vector in terms of the cylindrical coordinates. For example, if you have the vector \overline{A}=x^2\hat{i}+xy^2\hat{j}, then its cylindrical components would be:\overline{A}=r^2\cos^2(\theta)\hat{\rho} + r^2\sin(\theta)\cos(\theta)y^2\hat{\phi} + r\sin^2(\theta)y^2\hat{z}

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

A line integral of a vector field is a mathematical concept used in vector calculus to measure the total effect of a vector field along a specific path or curve.

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

A line integral of a vector field is calculated by evaluating the dot product of the vector field and the tangent vector of the path, and integrating this over the entire path.

3. What is the significance of a line integral of a vector field?

The line integral of a vector field can represent physical quantities such as work, force, and torque in real-world applications. It also allows for the calculation of the flux of a vector field through a closed curve.

4. What is the difference between a line integral and a path integral?

A line integral and a path integral are two terms used interchangeably to describe the same concept. Both refer to the integration of a vector field along a specified path or curve.

5. In what situations is the concept of a line integral of a vector field useful?

The line integral of a vector field is commonly used in physics and engineering to calculate quantities such as work, force, and torque. It is also used in fluid mechanics to calculate the flux of a vector field through a closed curve.

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