Path Integral of F = (x, y2, 2z) from a to b - Calculating Line Integrals

In summary, the problem asks us to evaluate two path integrals along a line from point a to point b. The first integral involves the vector field F, which is given as F = (x, y^2, 2z), and the differential vector dr. The second integral is a similar problem, but with the differential vector ds. To solve these integrals, we can use the path r(s) = (s, s, s) from (0,0,0) to (1,1,1), with s ranging from 0 to 1. By finding the appropriate vectors and integrating them, we can evaluate the two integrals.
  • #1
Prologue
185
1

Homework Statement


For F = (x, y2, 2z), evaluate the path integrals along the line of a to b:

[tex]\vec{a}=(0,0,0), \vec{b}=(1,1,1), \int^{b}_{a} \vec{F} \times d\vec{r}[/tex]

[tex]\int^{b}_{a} \vec{F} ds[/tex]

Homework Equations



No idea.

The Attempt at a Solution



I don't have a clue what these integrals even evaluate to. The first one should be a vector, and I have no idea what that even means, an integral that isn't a scalar. The second one is the same problem.
 
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  • #2
Write down the path r(s)=(s,s,s) from (0,0,0) to (1,1,1) with s going from 0 to 1. dr is (dr/ds)*ds which is (1,1,1)*ds. The integral of Fds is F(r(s))*ds. To integrate the vector ds just find the vector whose components are the integral of each component of the vector. I.e. (integral sds, integral s^2ds, integral 2sds). To find Fxdr cross the vector F(r(s)) with (1,1,1) and integrate that vector ds.
 

1. What is the meaning of "Integral of F cross dr"?

The "Integral of F cross dr" is a mathematical notation commonly used in physics and engineering to represent an integral over a path or line. It is also known as the line integral of a vector field.

2. How is "Integral of F cross dr" calculated?

The calculation of "Integral of F cross dr" involves integrating a vector field F along a specified path or line. This can be done by breaking the path into small segments and calculating the dot product of F with the infinitesimal displacement vector on each segment. The sum of these dot products is then integrated over the entire path to get the final value.

3. What are the applications of "Integral of F cross dr"?

"Integral of F cross dr" has various applications in physics and engineering, particularly in the fields of electromagnetism, fluid dynamics, and mechanics. It is used to calculate work done, potential energy, and other physical quantities along a specified path.

4. What is the difference between "Integral of F cross dr" and "Integral of F dot dr"?

The main difference between "Integral of F cross dr" and "Integral of F dot dr" lies in the direction of the vectors involved. "Integral of F cross dr" involves a vector product (also known as cross product) between F and dr, while "Integral of F dot dr" involves a scalar product (also known as dot product) between F and dr.

5. Are there any special cases of "Integral of F cross dr"?

Yes, there are two special cases of "Integral of F cross dr" - the path-independent integral and the path-dependent integral. In the path-independent case, the value of the integral remains constant regardless of the path taken, while in the path-dependent case, the value changes depending on the path.

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