Multivariable class, we'll be starting curvature

In summary, next week in my multivariable class, we'll be starting on curvature. As a self-proclaimed nerd, I decided to study it ahead of time but found it quite difficult. The equation for curvature is \kappa=|d\phi/ds|, where \phi is the angle between the curve's tangent vector and the horizontal, and s is the arc length. However, making it \kappa=|d\phi/dt/(ds/dt)| can be confusing, especially when trying to find an equation for \phi. A helpful tip is to use the equation \kappa=\frac{|\mathbf{r}'\times\mathbf{r}''|}{|\mathbf{
  • #1
FluxCapacitator
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Next week in my multivariable class, we'll be starting curvature, and, nerd that I am, I looked ahead to learn it ahead of time. I can usually at least understand the basics of a new concpet by myself, but curvature really threw me off. Maybe my brain's not right for it, maybe the book sucks, but I know my teacher sucks, so I'm pretty much going to have to learn it myself.

I know that curvature is [itex]\kappa=|d\phi/ds|[/itex], where [itex]\phi[/itex] is the angle between the curve's tangent vector and the horizontal, and s is the arc length.

I also get that the way to do this is to make it [itex]\kappa=|d\phi/dt/(ds/dt)|[/itex], I get lost, however, in actually finding a good equation for [itex]\phi[/itex].

Does anyone have any tips, resources, or advice?
 
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  • #2
From what I've learned, that equation is hard to work with. Have you seen it as this?:

[tex]\kappa=\frac{|\mathbf{r}'\times\mathbf{r}''|}{|\mathbf{r}'|^{3}}[/tex]

It's much easier to work with (r is the position vector). To show that the two definitions are equal, use the following fact:

[tex]\mathbf{r'}=\frac{ds}{dt}\mathbf{T}[/tex]
 
  • #3
That's a lot better :D . Thanks! That actually makes sense in a twisted sort of way, and it's a lot easier to use.
 

1. What is a multivariable class?

A multivariable class is a type of mathematical class that deals with functions of more than one variable. This means that instead of just having one independent variable, there are multiple independent variables that can affect the output of the function.

2. Why is it important to study curvature?

Curvature is an important concept in mathematics and physics because it measures how much a curve deviates from being a straight line. It has applications in fields such as engineering, computer graphics, and general relativity.

3. What topics are typically covered in a multivariable class?

In a multivariable class, you can expect to learn about concepts such as partial derivatives, multiple integrals, vector fields, and gradient, divergence, and curl. You may also cover topics like line and surface integrals, Green's theorem, and Stokes' theorem.

4. How does curvature relate to the study of surfaces?

The concept of curvature is closely related to the study of surfaces. Curvature measures the amount of bending or deviation from a flat surface at a given point. By studying curvature, we can understand the geometry of surfaces and how they behave in different scenarios.

5. What are some real-life applications of multivariable calculus and curvature?

Multivariable calculus and curvature have many real-life applications, such as in the field of computer graphics for creating 3D images and animations. They are also used in physics and engineering for understanding the behavior of objects in 3D space. Additionally, these concepts are essential in the study of fluid dynamics, electromagnetism, and general relativity.

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