Understanding Function Notation in Higher Level Mathematics

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SUMMARY

This discussion centers on understanding function notation in higher level mathematics, particularly in the context of three-dimensional functions like elliptic paraboloids. The example provided is f(x,y) = z = x² + y², illustrating how functions can represent surfaces in three-dimensional space. The conversation also touches on Newton's Second Law as a differential equation, raising questions about the dimensionality of force as a function of velocity and time. The distinction between the dimensionality of graphs and the functions they represent is clarified, emphasizing that while the graph may be two-dimensional, it exists within a three-dimensional context.

PREREQUISITES
  • Understanding of basic calculus concepts, including functions and derivatives.
  • Familiarity with three-dimensional coordinate systems.
  • Knowledge of differential equations, particularly in relation to Newton's Second Law.
  • Experience with graphing functions in multiple dimensions.
NEXT STEPS
  • Study the properties of elliptic paraboloids and their applications in higher mathematics.
  • Learn about the representation of functions in multiple dimensions, focusing on n-dimensional functions.
  • Explore differential equations in depth, particularly their applications in physics and engineering.
  • Investigate the relationship between force, velocity, and time in the context of Newtonian mechanics.
USEFUL FOR

Students in higher level mathematics, particularly those studying calculus, differential equations, and physics. This discussion is beneficial for anyone looking to deepen their understanding of function notation and its applications in engineering mathematics.

Artimodes
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Hello everyone,

I'm a newcomer to this forum! I've been having particular difficulty lately with understanding function notation in higher level mathematics. I felt like it was a general post so I posted in the general math section. I've finished calculus 3 and am going to take engineering math (basically ODE and applications) next spring.

It is my understanding that for a three dimensional function, for example an elliptic paraboloid, it's written like:

f(x,y)= z =x2+y2

However, after looking at Paul's Online Notes on ODE here, he starts to describe Newton's Second Law as a differential equation. I knew that, but his notation threw me off:

eq0006MP.gif


Is force in this case a three dimensional function, due to force being a function of the derivative of velocity while velocity is a function of time? Just confused.

Thanks for any help
 
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I don't know if it is helpful to refer to the function as being 3-dimensional , or functions

being n-dimensional. Strictly speaking, the graph is 2-dimensional, but it lives

in 3-dimensional space. In the sense of F(t,v) , you can say that F depends on two

parameters, and, if you were to graph force using axes for t,v the (two-dimensional)

graph would live in 3-D.
 

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