Vector calculus and force field

• tibphysic
In summary, The force field acting on a two-dimensional linear oscillator can be described by F = -xkx - yky. The work done moving against this force field from (1, 1) to (4, 4) can be compared by taking different straight-line paths. These paths include going from (1, 1) to (4, 1) to (4, 4), (1, 1) to (1, 4) to (4, 4), and (1, 1) to (4, 4) along the line x = y. The notation used for unit vectors may be unfamiliar, but the problem can be simplified if the force field is conservative.
tibphysic
1.10.1 The force field acting on a two-dimensional linear oscillator may be described by
F=−ˆxkx − ˆyky.
Compare the work done moving against this force field when going from (1, 1) to (4, 4)
by the following straight-line paths:
(a) (1, 1)→(4, 1)→(4, 4)
(b) (1, 1)→(1, 4)→(4, 4)
(c) (1, 1)→(4, 4) along x = y.
This means evaluating

(4,4)
(1,1)
F · dr
along each path.

hi tibphysic, welcome to pf, the idea is to have a try

do you know about conservative vector fields?

yes i do.. conservative vectors fields does not depend on the path taken.. just on the initial and the final point.

could be useful...

tibphysic said:
1.10.1 The force field acting on a two-dimensional linear oscillator may be described by
F=−ˆxkx − ˆyky.

I'm not familiar with that notation. Is that <-kx, -ky> or something else? Are ^x and ^y notations for i and j?

lanedance said:
could be useful...

True, but perhaps the intent of the exercise is to actually work them...

im assuming his notation ^x , read x hat. where the carrot should be over the x.
is equivalent to i , j , k . they use x hat and y hat and z hat in physics more.

For instance:

Along the path (1, 1)→(4, 1), $d\hat{r}=\hat{x}\,dx\,.$

So, find $\displaystyle\int_{(1,1)}^{(4,1)}\,dW=\int_{x=1}^{x=4}{\vec{F}\,|_{y=1}\cdot\hat{x}}\,dx$

Last edited:
yeah I thought the notation was a little strange but took them as unit vectors, so the potentials were of harmonic oscillator form

SammyS said:
For instance:

Along the path (1, 1)→(4, 1), $d\hat{r}=\hat{x}\,dx\,.$

So, find $\displaystyle\int_{(1,1)}^{(4,1)}\,dW=\int_{x=1}^{x=4}{\vec{F}\,|_{y=1}\cdot\hat{x}}\,dx$
True, but if the force field is conservative, as lanedance suggested in the first response, the problem is much easier. You don't need to do all those integrals.

1. What is vector calculus?

Vector calculus is a branch of mathematics that deals with the study of vector fields, which are quantities that have both magnitude and direction. It involves the use of mathematical operations such as differentiation and integration to study the properties and behavior of vector fields.

2. How is vector calculus used in physics?

Vector calculus is used extensively in physics, particularly in the study of force fields. It is used to describe and analyze the behavior of forces and other vector quantities, such as electric and magnetic fields. It also plays a crucial role in the study of motion and fluid dynamics.

3. What is a force field?

A force field is a region of space in which a force acts on an object. This force can be described by a vector quantity, which has both magnitude and direction. Examples of force fields include gravitational and electric fields.

4. How are force fields represented in vector calculus?

Force fields are represented as vector fields in vector calculus. This means that at each point in space, there is a vector that represents the magnitude and direction of the force acting on an object at that point. By using mathematical operations, such as taking the derivative, we can analyze the behavior of these force fields.

5. What are some real-world applications of vector calculus and force fields?

Vector calculus and force fields have numerous applications in various fields, including physics, engineering, and economics. Some specific examples include the analysis of fluid flow in pipes, the calculation of electric and magnetic fields in electronic devices, and the optimization of economic models. They are also used in computer graphics and animation to simulate realistic movement and interactions between objects.

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