Tangent Planes: Existence & Extensibility

Click For Summary
SUMMARY

In the discussion, participants explore the existence of tangent planes for functions of two variables, specifically z=f(x,y). It is established that most functions of two variables do indeed have tangent planes at any given point (x,y). The conversation also touches on the concept of extending this idea to higher dimensions, proposing the existence of tangent cubes and hypercubes. A reference to MathWorld's page on tangent planes is provided for further reading.

PREREQUISITES
  • Understanding of multivariable calculus
  • Familiarity with the concept of tangent lines
  • Knowledge of functions of two variables
  • Basic grasp of higher-dimensional geometry
NEXT STEPS
  • Research the mathematical definition and properties of tangent planes in multivariable calculus
  • Explore the concept of differentiability for functions of two variables
  • Learn about the extension of tangent planes to higher dimensions, including tangent cubes and hypercubes
  • Review resources on multivariable optimization techniques
USEFUL FOR

Students and professionals in mathematics, particularly those studying calculus and geometry, as well as educators looking to enhance their understanding of multivariable functions and their properties.

Char. Limit
Gold Member
Messages
1,222
Reaction score
23
Most functions y=f(x) have tangent lines for any point x.

Does a function z=f(x,y) have a tangent plane for any point x,y?

And could you extend this to higher dimensions if necessary? (Tangent cubes? Tangent hypercubes?)

Edit: Sorry, I thought I was posting in the General Math forum. Movement would be helpful, please.
 
Physics news on Phys.org
Yes. Check out -http ://mathworld.wolfram.com/TangentPlane.html
 

Similar threads

Can a Function Have Two Different Tangent Lines at the Same Point?
  • · Replies 2 ·
Replies
2
Views
2K
Find the equation of the tangent plane and normal to a surface
  • · Replies 1 ·
Replies
1
Views
1K
Undergrad Approximation of a function of two variables
  • · Replies 3 ·
Replies
3
Views
2K
High School What Is the Fourth Dimension and How Does It Work?
  • · Replies 3 ·
Replies
3
Views
3K
Lagrange multipliers understanding
  • · Replies 8 ·
Replies
8
Views
2K
High School Tangents to a curve as functions of different variables
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Gradient Vectors: Perpendicular to Level Curves
  • · Replies 4 ·
Replies
4
Views
2K
Find intersections between sphere and parallel tangent planes
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad Question about gradient, tangent plane and normal line
  • · Replies 1 ·
Replies
1
Views
2K
MHB Cartesian equation of the tangent plane
  • · Replies 10 ·
Replies
10
Views
3K