Tangent Planes: Existence & Extensibility

In summary, a tangent plane is a flat surface that touches a curved surface at a single point and is used to approximate the behavior of the curved surface at that specific point. Its existence is determined by the differentiability of the function at that point and is significant in mathematics for understanding the behavior of curved surfaces. It can be extended to cover the entire curved surface if the function is differentiable at every point, and is commonly used in real-world applications such as engineering, computer graphics, and physics.
  • #1
Char. Limit
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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.
 
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  • #2
Yes. Check out -http ://mathworld.wolfram.com/TangentPlane.html
 

1. What is a tangent plane?

A tangent plane is a flat surface that touches a curved surface at a single point. It is used to approximate the behavior of the curved surface at that specific point.

2. How is the existence of a tangent plane determined?

The existence of a tangent plane is determined by the differentiability of the function at that point. If a function is differentiable at a point, it means that the tangent plane exists at that point.

3. What is the significance of tangent planes in mathematics?

Tangent planes are important in mathematics because they help us understand the behavior of curved surfaces. They allow us to approximate the slope and direction of a surface at a specific point, which is useful in many applications such as optimization and physics.

4. Can a tangent plane be extended to cover the entire curved surface?

Yes, a tangent plane can be extended to cover the entire curved surface if the function is differentiable at every point on the surface. This is known as the extensibility of the tangent plane.

5. How are tangent planes used in real-world applications?

Tangent planes have many practical applications, such as in engineering, computer graphics, and physics. They are used to approximate the behavior of curved surfaces in these fields, making calculations and predictions more accurate and efficient.

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