Parametric equations and derivatives

In summary, the formula for calculating df/dt, or the total derivative of a function with multiple variables, is given by adding the partial changes caused by the individual variables (x, y, and z) as they change with respect to t. This is represented by the use of "round d" (∂) for partial changes and "regular d" (d) for non-partial changes.
  • #1
Shaybay92
124
0
Just a quick question... if we have f(x,y,z) and x(t), y(t), z(t), without substituting in what x y and z are in f, how do we calculate df/dt?
 
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  • #2
Shaybay92 said:
Just a quick question... if we have f(x,y,z) and x(t), y(t), z(t), without substituting in what x y and z are in f, how do we calculate df/dt?

[tex]\frac {df} {dt} = \frac {\partial f} {\partial x} \frac {dx} {dt} + \frac {\partial f} {\partial y} \frac {dy} {dt} + \frac {\partial f} {\partial z} \frac {dz} {dt}[/tex]
 
  • #3
Where did this come from? I can't see why we should add the contributions of each?
 
  • #4
Shaybay92 said:
Where did this come from? I can't see why we should add the contributions of each?

This is the standard formula for multiple-variable derivatives, also called the "total derivative".

In words: if t would increase a little bit, x, y, and z each will change a little bit as well.
This will make f change a little bit as well. There will be a partial change caused by the change in x, and also a partial changed caused by the change in y.
All in all, all the partial changes need to be added.

This is the reason the "round d" is used in ∂f/∂x to signify it's about the partial change of f, due to a change in x. This needs to be multiplied by the change that x takes due to the change in t. Since the last is not a "partial" change, a "regular d" is used as in dx/dt.
 
  • #5
Thankyou!
 

1. What are parametric equations?

Parametric equations are a set of equations that express the coordinates of a point in terms of one or more independent variables, usually denoted by t. These equations are commonly used to represent curves or surfaces in mathematics and physics.

2. How do I find the derivative of a parametric equation?

To find the derivative of a parametric equation, you can use the chain rule. First, take the derivative of each individual equation with respect to the independent variable t. Then, substitute these derivatives into the formula for the chain rule: dy/dx = (dy/dt) / (dx/dt).

3. What is the difference between a parametric equation and a Cartesian equation?

A parametric equation describes a curve or surface in terms of one or more independent variables, while a Cartesian equation describes the same curve or surface in terms of x and y coordinates. Parametric equations are often used when working with more complex curves, while Cartesian equations are more commonly used for simpler curves.

4. Can parametric equations be used to represent three-dimensional figures?

Yes, parametric equations can be used to represent three-dimensional figures by adding a third parameter, usually denoted by z. This allows for the representation of curves and surfaces in three-dimensional space.

5. What are some real-life applications of parametric equations?

Parametric equations are used in many fields, including physics, engineering, and computer graphics. They can be used to model the motion of objects, design complex shapes and curves, and generate computer-generated images and animations.

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