Function of 3 variable help

In summary, the conversation discusses a problem involving a 4 dimensional shape and its corresponding equations, including sphere, ellipse, and hyperbola. The speaker provides a solution for question one and two, describing the shape as a "collection of equally spaced concentric spheres" and "collection of unequally spaced concentric spheres" respectively. However, they are unsure about why using the radius of √w instead of w makes the sphere unequally spaced. The speaker mentions their teacher barely touched on the topic and suggests looking up the equations of conic sections in 3 dimensions to better visualize the shapes in 4 dimensions.
  • #1
170
2

Homework Statement



The problem is given in the picture attached. However we are given the equation of a 4 dimensional shape with certain choice given to explain the shape in that dimension.

Homework Equations



Equations of different shape such as:

sphere: √(x^2+y^2+z^2) = r
Ellipse: (x-h)/a^2 + (y-k)/b^2 = K
Hyperbola: (x-h)/a^2 - (y-k)/b^2 = 1

The Attempt at a Solution



Well for question one and two, I said 1 ( which is w=x^2+y^2+z^2) is a "a collection of equally spaced concentric spheres" and 2 ( w=√(x^2+y^2+z^2) was "a collection of unequally spaced concentric spheres ".Though I know the difference between both is the radius where one is w and the other is √w. But why does have a radius of √w make the sphere unequally spaced?

My teacher barely touched the topic so I have a hard time figuring out how to visualize these shapes.
 
Physics news on Phys.org
  • #2
I couldn't see my attachment... here it is ( trying again)
 

Attachments

  • untitled 2.jpg
    untitled 2.jpg
    26.1 KB · Views: 344
  • #3
You are correct about #1.
 
  • #4
You are correct about #1. I do know the answers to some of the others, and can guess some, but perhaps it would be best to tell you how I would go about this if I had to do the problem.

Go back to 3 dimensions, and look up the equations of the conic sections. Answer the questions when z is omitted. It has to be similar in 4 dimensions.
 

What is a function of 3 variables?

A function of 3 variables is a mathematical relationship where the output value is determined by three input values. It can be represented as f(x,y,z) = w, where x, y, and z are the input variables and w is the output value.

How do you graph a function of 3 variables?

A function of 3 variables can be graphed on a 3-dimensional coordinate system. The x and y axes represent the first two input variables, and the z axis represents the output value. The graph will show how the output value changes as the input values vary.

What is the domain and range of a function of 3 variables?

The domain of a function of 3 variables is the set of all possible input combinations. The range is the set of all possible output values. In other words, the domain is the values that can be plugged into the function, and the range is the resulting output values.

How do you find the critical points of a function of 3 variables?

The critical points of a function of 3 variables are the points where the partial derivatives of all three input variables are equal to zero. These points can be found by setting each partial derivative equal to zero and solving the resulting system of equations.

Can a function of 3 variables be optimized?

Yes, a function of 3 variables can be optimized by finding the critical points and determining which point gives the maximum or minimum output value. This can be done by using the second derivative test or by creating a contour map of the function.

Suggested for: Function of 3 variable help

Replies
10
Views
761
Replies
11
Views
1K
Replies
1
Views
516
Replies
3
Views
478
Replies
13
Views
281
Replies
4
Views
707
Replies
1
Views
389
Replies
27
Views
303
Replies
10
Views
847
Back
Top