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## Multivariable calculus

### Course: Multivariable calculus>Unit 3

Lesson 1: Tangent planes and local linearization

# What is a tangent plane

The "tangent plane" of the graph of a function is, well, a two-dimensional plane that is tangent to this graph. Here you can see what that looks like. Created by Grant Sanderson.

## Want to join the conversation?

• at , I don't understand how the point on a graph can be described purely by (x,y.) doesn't it need a z coordinate to be properly described as well? likewise, the example of a point on a line in R2 says that that point could be described as (x), but wouldn't a y coordinate be needed to properly show it's location? • It is a bit subtle. Because we assume the function is given, the idea is that "which point we are talking about" is completely determined by choosing (x, y). It's true that if you actually want to locate the final point in R3, then you need to plug in x and y and get f(x,y), which you could think of as the z-coordinate. But since choosing x and y is all the choice you are allowed, and that gives one and only one z, we can say that the point on the surface is picked out entirely once you have chosen (x,y).

Likewise in the one-dimensional case. For example, if f(x) = x², then we can say "find the tangent line at x = 3". Although you actually need to compute the y-value of 9 in order to give the final answer L(x) = 9 + 6(x - 3), still, the "point" was completely determined by "at x = 3", since the function f(x) = x² was given. It was not necessary to say "at the point (3, 9)".
• Would that mean people are just tangent objects of the space-time continuum? (When thinking fourth-dimensionally)
(1 vote) • Doubt it. The tangent to a 4 dimensional object would be a 3d surface. But, I would think the surface would be highly specific, as the tangent to a 2d graph is a straight line and only a straight line and the tangent to a 3d surface would be a flat plane and only a flat plane. Both the line and plane are infinite in length/size, and people are not "regular" in shape and certainly not infinite in size. Thus, I doubt people are just tangent objects.
• Can we say that the plane is the span of the vectors [∂f/∂x,0] (transposed) and [0,∂f/∂y] (transposed)?
(1 vote) • I think you are kinda of right but you have to keep in mind that we are in 3d dimensions so I think a more correct answer would be that the plane is the span of
{ [1, 0 ,∂f/∂x] , [0, 1, ∂f/∂y] }

where each vector is transposed.
Remember that the partial derivative with respect to x can be thought of as tangent slope for the curve generated by our graph and a xz plane.

I hope this helps. 