What do the symbols in the line integral equation mean?

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In summary: And this one, ds, is the "differential of arc length." The differential dX is the change in X; the derivative, dX/dt, is the speed. The differential of arc length, ds, is the change in arc length; the derivative of arc length, ds/dt, is the "speed" along the curve.In summary, the "ds" notation in the line integral formula indicates the variable of integration, which can be changed to any other variable. The \gamma represents a parametrization of the curve, which is necessary for the integral. A parametrization is a way to express a curve in terms of one variable, and the "ds" in the integral is the differential of
  • #1
nayfie
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Hey guys. I've just had a few lectures on line integrals. My lecturer has told me the following:

[itex]\int_{C}^{} f(s) \text{d}s = \int_{a}^{b} f(\gamma(t))\|\gamma'(t)\| \text{d}t[/itex]

Unfortunately he hasn't explained this topic very well. I understand what's going on with a line integral but have these few questions:

- What is the 'ds' on the left hand side?
- How do we determine what gamma is?
 
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  • #2
The "ds" is just a notation to indicate the variable towards we integrate. You can change it by dt or du or whatever (although ds seems like the canonical choice).

The [itex]\gamma[/itex] is a parametrization of the curve. This is usually given, although not always.

For example, if C is a circle, then

[itex]\gamma:[0,2\pi[\rightarrow \mathbb{R}^2:t\rightarrow (\cos(t),\sin(t))[/itex]

This is one of the many possible parametrizations of the circle.
Note that the integral does depend on the parametrization used: for example, if we decide to run C in the other direction, then the integral can turn up to be negative. Thus C should always come with some kind of orientation if we want to find a right parametrization.
 
  • #3
Firstly, thanks for the reply.

Can you describe what you mean by a parametrization? In your example you've used a range, but in an example we did in class the lecturer used the function of the curve.

I'm confused :(
 
  • #4
"ds" is the "differential of arc length". A curve, being one dimensional, can be expressed in terms of one variable. For example, a curve in 3 dimensional space, given an ordinary xyz-coordinate system, can be written x= f(t), y= g(t), z= h(t). If you like, you think of t as "time" and those three equations give the position of an object moving along that curve.

If we have two points, say [itex](x_0, y_0, z_0)[/itex] and [itex](x_1, y_1, z_1)[/itex], then the straight line distance between the points is [itex]\sqrt{(x_1-x_0)^2+ (y_1-y_0)^2+ (z_1-z_0)^2}[/itex], from the Pythagorean theorem. For a curve, do the usual reduction to "differentials" or derivatives that you have surely seen in Calculus to get the "differential of arc length", [itex]ds= \sqrt{dx^2+ dy^2+ dx^2}[/itex] or, in terms of the parameter, t, [itex]dx= \sqrt{(dx/dt)^2+ (dy/dt)^2+ (dz/dt)^2}dt[/itex].

IF the curve, in two dimensions, is given as a function, y= f(x), you can use x itself as parameter. Formally, that would be "x= t, y= f(t)" so that the differential of arc length would be [itex]ds= \sqrt{(dx/dt)^2+ (dy/dt)^2}dt= \sqrt{1+ (df/dt)^2}dt[/itex] or, less formally, [itex]ds= \sqrt{(1+ (dy/dx)^2}dx[/itex].
 
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  • #5
Not sure about the second part but the "ds" basically means the derivative of variable "s." This vairable could have been any other letters, such as x - in that case, the derivate of "x" would be stated as "dx."
 
  • #6
CallMeShady said:
Not sure about the second part but the "ds" basically means the derivative of variable "s." This vairable could have been any other letters, such as x - in that case, the derivate of "x" would be stated as "dx."
You mean differential, not derivative. Derivatives are things like dy/dx, and dx and dy are called differentials.
 

1. What is a line/curve integral?

A line/curve integral is a mathematical concept used in calculus to determine the area under a curve or the length of a curve. It involves breaking down a curve into small segments, calculating the area or length of each segment, and then summing them up to get the total area or length.

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

A line integral is used to calculate the area under a curve in a two-dimensional plane, whereas a curve integral is used to calculate the area under a curve in a three-dimensional space. The main difference is the number of dimensions in which the curve is being evaluated.

3. What are some real-world applications of line/curve integrals?

Line/curve integrals have many real-world applications, such as in physics for calculating work done by a force, in engineering for determining the center of mass of an object, in economics for calculating consumer surplus, and in computer graphics for creating smooth curves and surfaces.

4. What are some techniques for solving line/curve integral problems?

There are several techniques for solving line/curve integral problems, such as using the fundamental theorem of calculus, substitution, integration by parts, and trigonometric identities. It is important to understand the properties of integrals and have a strong foundation in calculus to effectively solve these problems.

5. Can line/curve integrals have negative values?

Yes, line/curve integrals can have negative values. This can occur when the curve being evaluated is below the x-axis or when the function being integrated has negative values. The negative value indicates that the area or length is below the x-axis or is in the opposite direction of the chosen orientation for the curve.

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