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Chapter 2: Problem 3

Show that the equation of the tangent plane of a surface which is the graph of a differentiable function \(z=f(x, y)\), at the point \(p_{0}=\left(x_{0}, y_{0}\right)\), is given by $$ z=f\left(x_{0}, y_{0}\right)+f_{x}\left(x_{0}, y_{0}\right)\left(x-x_{0}\right)+f_{y}\left(x_{0}, y_{0}\right)\left(y-y_{0}\right) $$ Recall the definition of the differential \(d f\) of a function \(f: R^{2} \rightarrow R\) and show that the tangent plane is the graph of the differential \(d f_{p}\).

Short Answer

Expert verified
The equation of the tangent plane is \(z = f(x_{0}, y_{0}) + f_{x}(x_{0}, y_{0})(x - x_{0}) + f_{y}(x_{0}, y_{0})(y - y_{0})\).

Step by step solution

01

Identify the given surface and point

Consider the surface given by the differentiable function \(z = f(x, y)\). The point at which the tangent plane is to be found is \(p_{0} = (x_{0}, y_{0})\).
02

Recall the definition of the differential

The differential \(df\) of a function \(f : \mathbb{R}^{2} \rightarrow \mathbb{R}\) at a point \( (x_{0}, y_{0}) \) is given by: \[ df_{(x_{0}, y_{0})} = f_{x}(x_{0}, y_{0}) dx + f_{y}(x_{0}, y_{0}) dy \]
03

Express the tangent plane equation

The tangent plane to the surface \(z = f(x, y)\) at the point \((x_{0}, y_{0}, f(x_{0}, y_{0}))\) is given by approximating \(f(x, y)\) using the differential. This leads to: \[ z \approx f(x_{0}, y_{0}) + f_{x}(x_{0}, y_{0})(x - x_{0}) + f_{y}(x_{0}, y_{0})(y - y_{0}) \]
04

Identify the differential in the equation

Observe that the expression \( f(x_{0}, y_{0}) + f_{x}(x_{0}, y_{0})(x - x_{0}) + f_{y}(x_{0}, y_{0})(y - y_{0}) \) directly corresponds to the differential \( df_{(x_{0}, y_{0})} \).
05

Finalize the tangent plane equation

Thus, the equation of the tangent plane to the surface at the point \((x_{0}, y_{0})\) is: \[ z = f(x_{0}, y_{0}) + f_{x}(x_{0}, y_{0})(x - x_{0}) + f_{y}(x_{0}, y_{0})(y - y_{0}) \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Differential Function
A differential function, often denoted as \( f: \mathbb{R}^{2} \rightarrow \mathbb{R} \), forms the backbone of many concepts in calculus and differential geometry. It describes how a function changes as its input values change. To better understand, we look at the concept of a differential, denoted by \( df \). The differential \( df \) of a function \( f \) at a particular point \( (x_{0}, y_{0}) \) can be written as:
\[ df_{(x_{0}, y_{0})} = f_{x}(x_{0}, y_{0}) \ dx + f_{y}(x_{0}, y_{0}) \ dy \]
Interpretation:
  • \( f_{x}(x_{0}, y_{0}) \) is the partial derivative of \( f \) with respect to \( x \) at point \( (x_{0}, y_{0}) \).
  • \( f_{y}(x_{0}, y_{0}) \) is the partial derivative of \( f \) with respect to \( y \) at point \( (x_{0}, y_{0}) \).
Essentially, the differential approximates how much the output \( f(x,y) \) changes due to small changes in \( x \) and \( y \). This is crucial for deriving equations like the tangent plane.
Differential Geometry
Differential geometry combines concepts from calculus and geometry to study shapes and surfaces. A primary tool in this field is the differential, which helps us understand and compute things like the tangent plane to a surface.
Consider a surface described by \( z = f(x, y) \). Differential geometry examines how this surface behaves locally, especially around points of interest like \( (x_{0}, y_{0}, z_{0}) \). One key result from differential geometry is that the tangent plane at any point on a surface can be approximated using the differential function. For the surface \( z = f(x, y) \), at point \( (x_{0}, y_{0}) \), the equation of the tangent plane is derived using the first derivatives, similar to how we use the differential to approximate changes in function values.
Tangent Plane
The tangent plane is a flat, two-dimensional plane that just 'touches' a surface precisely at one point without cutting into it. This concept is vital in multivariable calculus and geometry.
To find the equation of the tangent plane to a surface given by \( z = f(x, y) \) at a point \( (x_{0}, y_{0}) \), we use an approximation based on the differential. The equation is given by:
\[ z = f(x_{0}, y_{0}) + f_{x}(x_{0}, y_{0})(x - x_{0}) + f_{y}(x_{0}, y_{0})(y - y_{0}) \]
This expression shows how the surface's height \( z \) changes linearly around the point \( (x_{0}, y_{0}) \). Here:
  • \( f(x_{0}, y_{0}) \) is the height of the surface at \( (x_{0}, y_{0}) \).
  • \( f_{x}(x_{0}, y_{0}) \) and \( f_{y}(x_{0}, y_{0}) \) are the slopes of the surface in the \( x \) and \( y \) directions, respectively.
By combining these, we can predict the behavior and position of the surface near the point \( (x_{0}, y_{0}) \) using the tangent plane.

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Most popular questions from this chapter

Show that the perpendicular projections of the center \((0,0,0)\) of the ellipsoid $$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1 $$ onto its tangent planes constitute a regular surface given by $$ \left\\{(x, y, z) \in R^{3} ;\left(x^{2}+y^{2}+z^{2}\right)^{2}=a^{2} x^{2}+b^{2} y^{2}+c^{2} z^{2}\right\\}-\\{(0,0,0)\\} $$

Let \(R^{2}=\left\\{(x, y, z) \in R^{3} ; z=-1\right\\}\) be identified with the complex plane \(\mathbb{C}\) by setting \((x, y,-1)=x+i y=\zeta \in \mathbb{C}\). Let \(P: \mathbb{C} \rightarrow \mathbb{C}\) be the complex polynomial $$ P(\zeta)=a_{0} \zeta^{n}+a_{1} \zeta^{n-1}+\cdots+a_{n}, \quad a_{0} \neq 0, a_{i} \in \mathbb{C}, i=0, \ldots, n $$ Denote by \(\pi_{N}\) the stereographic projection of \(S^{2}=\left\\{(x, y, z) \in R^{3}\right.\); \(\left.x^{2}+y^{2}+z^{2}=1\right\\}\) from the north pole \(N=(0,0,1)\) onto \(R^{2}\). Prove that the map \(F: S^{2} \rightarrow S^{2}\) given by $$ \begin{aligned} &F(p)=\pi_{N}^{-1} \circ P \circ \pi_{N}(p), \quad \text { if } p \in S^{2}-\\{N\\} \\ &F(N)=N \end{aligned} $$ is differentiable.

(Theory of Contact.) Two regular surfaces, \(S\) and \(\bar{S}\), in \(R^{3}\), which have a point \(p\) in common, are said to have contact of order \(\geq 1\) at \(p\) if there exist parametrizations with the same domain \(\mathbf{x}(u, v), \overline{\mathbf{x}}(u, v)\) at \(p\) of \(S\) and \(\bar{S}\), respectively, such that \(\mathbf{x}_{u}=\overline{\mathbf{x}}_{u}\) and \(\mathbf{x}_{v}=\overline{\mathbf{x}}_{v}\) at \(p .\) If, moreover, some of the second partial derivatives are different at \(p\), the contact is said to be of order exactly equal to 1 . Prove that a. The tangent plane \(T_{p}(S)\) of a regular surface \(S\) at the point \(p\) has contact of order \(\geq 1\) with the surface at \(p\). b. If a plane has contact of order \(\geq 1\) with a surface \(S\) at \(p\), then this plane coincides with the tangent plane to \(S\) at \(p\). c. Two regular surfaces have contact of order \(\geq 1\) if and only if they have a common tangent plane at \(p\), i.e., they are tangent at \(p\). d. If two regular surfaces \(S\) and \(\bar{S}\) of \(R^{3}\) have contact of order \(\geq 1\) at \(p\) and if \(F: R^{3} \rightarrow R^{3}\) is a diffeomorphism of \(R^{3}\), then the images \(F(S)\) and \(F(\bar{S})\) are regular surfaces which have contact of order \(\geq 1\) at \(f(p)\) (that is, the notion of contact of order \(\geq 1\) is invariant under diffeomorphisms). e. If two surfaces have contact of order \(\geq 1\) at \(p\), then \(\lim _{r \rightarrow 0}(d / r)=0\), where \(d\) is the length of the segment which is determined by the intersections with the surfaces of some parallel to the common normal, at a distance \(r\) from this normal.

Show that the paraboloid \(z=x^{2}+y^{2}\) is diffeomorphic to a plane.

Let \(f(x, y, z)=(x+y+z-1)^{2}\). a. Locate the critical points and critical values of \(f\). b. For what values of \(c\) is the set \(f(x, y, z)=c\) a regular surface? c. Answer the questions of parts a and b for the function \(f(x, y, z)=\) \(x y z^{2}\).
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