Parameterised a function

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In summary, there is no one specific method for parameterizing a function of scalar and vector. It depends on the specific case and some familiar relations between x and y can be exploited. There are also an infinite number of different parameterizations for any curve. Some functions may not be able to be parameterized.
  • #1
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Is there a specific method to parameterised a function of scalar and vector?

I was reading through this website but then, can't understand how it read taht parameterised steps.

Any guide?

(http://www.math.umn.edu/~nykamp/m2374/readings/stokesex/index.html
under double check example)

Thanks
 
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  • #2
I don't think there is a method that works in all cases. But when the curve is specified as a level curve f(x,y)=0, you can try to solve for y. In that case, just use x as the parameter.

Exemple: Consider the level curve C={(x,y): y-x²-2x=0. We can solve of y like so: y=x²+2x. In that case, the parametrized curve is r(t)=(t²+2t,t), t [itex]\in \mathbb{R}[/itex].

In other cases, if the relation between x and y is something familiar you can try to exploit this as illustrated in the following exemple:

Exemple: Consider the level curve C={(x,y): (x/a)²-(y/b)²=1}. This resembles the identity cosh²(t)-sinh²(t)=1. So set x(t)/a=cosh(t) and y(t)/b=sinh(t), i.e. let r(t)=(acosh(t),bsinh(t)) t [itex]\in \mathbb{R}[/itex] parametrize the level curve.

N.B. You can convince yourself that this parametrization covers the whole curve because given any value of x, there is a corresponding value of t for which acosh(t)=x and similarly for y. [cosh(t) is surjective on the x-axis and sinh(t) is surjective on the y axis]. See http://en.wikipedia.org/wiki/Image:Sinh_cosh_tanh.svg
 
  • #3
No wait, cosh(t) is not surjective on the real line, it only covers (1, infty). So r(t) would only cover part of C.
 
  • #4
As quasar987 said, there is no one method of parameterizing a curve. In fact, there exist an infinite number of different parameterizations for any curve.

For the example given, because the path (From the origin, (0, 0, 0) along the z-axis to (0, 0 1), then along the quarter circle to (0, 1, 0), then along the y-axis to (0, 0, 0) again) is not "smooth" (there are corners at (0,0,0), (0, 1, 0) and (0, 0, 1)), you would break it into three pieces.

First the line from (0, 0, 0) to (0, 0, 1). At any point on that line, x=y= 0
An obvious parameterization is to use z itself as parameter: x= 0, y= 0, z= t, with [itex]0\le t\le 1[/itex].
Second, the quarter circle from (0, 0, 1) to (0, 1, 0). At every point on that circle x= 0 and y2+ z2= 1. A "standard" parameterization for a circle is to use sine and cosine: x= 0, y= sin(t), z= cos(t). Then x= 0 always while y2+ z2= sin2(t)+ cos2(t)= 1 for all t. Of course, it is z= cos(t) and not y because when t= 0, z= cos(0)= 1 and y= sin(0)= 0 as required. When t= [itex]\pi/2[/itex], z= cos([itex]\pi/2[/itex])= 0 and y= sin([itex]\pi/2[/itex])= 1 so [itex]0\le t\le \pi/2[/itex].

Finally, the line from (0, 1, 0) to (0, 0, 0). Obviously x= z= 0 at every point. We could use y itself as parameter: x= 0, y= t, z= 0 with t going from 1 to 0. Another possibility is x= 0, y= 1- t, z= 0 with t going from 0 to 1.
 
  • #5
Oh rite, i see.
Thanks for helps!

Edit: oh, btw, is there any functions that can't be parameterized?
 
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1. What is a parameterised function?

A parameterised function is a type of function in which one or more variables, known as parameters, are used to represent a set of values. These values can then be passed into the function when it is called, allowing the function to perform different operations based on the input.

2. How do you declare a parameterised function?

A parameterised function is declared in the same way as a regular function, but with the addition of one or more parameters inside the parentheses after the function name. For example: function myFunction(param1, param2) { // code block }

3. What is the purpose of using parameters in a function?

The use of parameters in a function allows for more flexibility and reusability. By passing different values into the function, it can perform different operations without having to create multiple versions of the same function. This makes code more efficient and easier to maintain.

4. Can a parameterised function have a return value?

Yes, a parameterised function can have a return value just like a regular function. The parameters passed into the function can be used in the function's code to calculate a result, which can then be returned using the return keyword.

5. Can a function have both parameters and arguments?

Yes, a function can have both parameters and arguments. Parameters are used in the function declaration, while arguments are the actual values passed into the function when it is called. The number of arguments passed in must match the number of parameters declared in the function.

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