Question

Consider the function \(F(x, y, z)=0,\) which is differentiable at \(P\left(x_{0}, y_{0}, z_{0}\right)\). Give the definition of the tangent plane at \(P\) and the normal line at \(P .\)

Short answer

The equation of the tangent plane at \(P\left(x_{0}, y_{0}, z_{0}\right)\) is given by: \[F_{x}\left(x_{0}, y_{0}, z_{0}\right)\left(x-x_{0}\right)+F_{y}\left(x_{0}, y_{0}, z_{0}\right)\left(y-y_{0}\right)+F_{z}\left(x_{0}, y_{0}, z_{0}\right)\left(z-z_{0}\right)=0\]And the normal line at \(P\) has the parametric equations: \[x=x_{0}+tF_{x}\left(x_{0}, y_{0}, z_{0}\right)\]\[y=y_{0}+tF_{y}\left(x_{0}, y_{0}, z_{0}\right)\]\[z=z_{0}+tF_{z}\left(x_{0}, y_{0}, z_{0}\right)\]

Step-by-step solution

Derive the Tangent Plane

The tangent plane is a linear approximation to the function near \(P\). Its equation can be given by the formula: \[F_{x}\left(x_{0}, y_{0}, z_{0}\right)\left(x-x_{0}\right)+F_{y}\left(x_{0}, y_{0}, z_{0}\right)\left(y-y_{0}\right)+F_{z}\left(x_{0}, y_{0}, z_{0}\right)\left(z-z_{0}\right)=0\] where \(F_{x}\), \(F_{y}\) and \(F_{z}\) are partial derivatives of \(F\) with respect to \(x\), \(y\) and \(z\), respectively.

Derive the Normal Line

The normal line at point \(P\) is a straight line perpendicular to the tangent plane. Its equation, in 3D space is given by the parametric equations: \[x=x_{0}+tF_{x}\left(x_{0}, y_{0}, z_{0}\right)\]\[y=y_{0}+tF_{y}\left(x_{0}, y_{0}, z_{0}\right)\]\[z=z_{0}+tF_{z}\left(x_{0}, y_{0}, z_{0}\right)\]where \(t\) is the parameter.

Key concepts

Differentiable Functions

In calculus, a differentiable function plays a crucial role in understanding the changing behavior of variables. When a function is differentiable at a point, like our function F at point P, it means that there is a smooth piece of curve at P, without any sharp turns or cusps.

Differentiability implies that the function can be locally approximated by a linear function, which is tangent to the curve at the point in question. Hence, for a function of several variables, differentiability at a point allows for the existence of a tangent plane to the surface represented by the function at that point.

To ensure a function is differentiable, we first look for continuities in the partial derivatives at the point of interest. If the function is not continuous, it cannot be smooth, and thus, it's not differentiable. By ensuring differentiability, we can confidently talk about the tangent plane and normal line with respect to the function at the given point P.

Partial Derivatives

The concept of partial derivatives comes into play when dealing with multivariable functions, like F(x, y, z) from our exercise. A partial derivative tells us how the function changes as one variable changes, while the others are held constant.

For instance, the notation F_x represents the partial derivative of F with respect to x, which quantifies the rate of change of the function value as x varies and y and z are fixed. Similarly, F_y and F_z measure changes along the y and z dimensions respectively.

In our exercise, these partial derivatives are used to describe the orientation of the tangent plane and the direction of the normal line at point P. They are essential in defining the geometry of surfaces in 3D space and are fundamental to understanding gradients, tangent planes, and normal lines.

Linear Approximation

The idea behind a linear approximation is to use a linear expression to approximate the values of a function near a certain point. Essentially, this linear expression is the equation of the tangent plane that we derived in the exercise.

The tangent plane at a point on a surface provides the best linear approximation of the surface at that spot. Just as a tangent line to a curve in 2D represents the curve at a point, the tangent plane in 3D touches the surface at a single point and extends out in a flat, two-dimensional plane.

Understanding Linear Approximation

Imagine smoothing out a curved surface so it's perfectly flat right around point P. This flat area is what we are estimating with the linear approximation. The accuracy of this approximation is best near point P and gets worse as you move away from P. It's akin to assuming the Earth is flat when you're taking a short walk — it feels flat where you are, but the further you go, the more you notice the curvature.

In our exercise, the coefficients of the linear terms in the tangent plane's equation come from the partial derivatives of F, which signifies how steeply the surface is sloping at that point. This concept of linear approximation is fundamental in various applications, such as optimization and numerical analysis.