Introduction to 3-D Surfaces

Welcome to the third dimension! Up until now, you’ve mostly dealt with 2-D graphs where \( y \) depends on \( x \). In this chapter, we explore Additional Pure Mathematics by looking at surfaces where the height, \( z \), depends on two different variables, \( x \) and \( y \). Imagine you are standing on a mountain: your elevation (\( z \)) depends on your position east-west (\( x \)) and north-south (\( y \)). That is a 3-D surface!

Don't worry if this seems a bit "out there" at first. We will use what you already know about coordinates and gradients and simply add one more layer to it. By the end of these notes, you'll be able to navigate these 3-D landscapes like a pro.


1. Understanding 3-D Surfaces

A 3-D surface can be defined in two ways:

  • Explicitly: \( z = f(x, y) \). This tells you exactly how to calculate the height \( z \) if you know \( x \) and \( y \). For example, \( z = x^2 + y^2 \).
  • Implicitly: \( g(x, y, z) = c \). Here, the variables are all mixed together. For example, the equation of a sphere: \( x^2 + y^2 + z^2 = 25 \).

Did you know? In Stage 2 of your course, you will encounter surfaces that involve more than just powers of \( x \) and \( y \). You'll see trigonometric functions, logarithms, and exponentials, such as \( z = e^x \sin(y) \). The rules remain the same, but the algebra gets a bit more exciting!

Quick Review:

If you see \( z = f(x, y) \), think of \( x \) and \( y \) as the floor of a room, and \( z \) as the distance to the ceiling at that specific spot.

Key Takeaway: A function of two variables describes a surface in 3-D space where every point \( (x, y) \) on a flat plane maps to a height \( z \).


2. Sections and Contours

Visualizing a 3-D shape on a 2-D piece of paper is hard. To help, we use two techniques: Sections and Contours.

Sections

A section is what you get if you "slice" the surface with a vertical plane.
Analogy: Slicing a loaf of bread. Each slice shows you the cross-section shape of the loaf at that point.

  • If we keep \( x \) constant (\( x = a \)), we get a section of the form \( z = f(a, y) \).
  • If we keep \( y \) constant (\( y = b \)), we get a section of the form \( z = f(x, b) \).

Contours

A contour is what you get if you "slice" the surface horizontally.
Analogy: Think of a topographic map used by hikers. The lines on the map are contours showing points of equal height.

To find a contour, we set \( z \) to a constant value \( c \). The equation becomes \( f(x, y) = c \). This gives us a 2-D curve on the \( xy \)-plane.

Key Takeaway: Sections slice vertically (showing side views), and contours slice horizontally (showing a bird's-eye view).


3. Partial Differentiation

This is the core tool of the chapter. Partial differentiation allows us to find the gradient of a surface in a specific direction.

The Golden Rule: When you differentiate with respect to one variable, treat the other variable as if it were just a number (a constant).

Notation

There are two common ways to write partial derivatives:

  • With respect to \( x \): \( \frac{\partial f}{\partial x} \) or \( f_x \)
  • With respect to \( y \): \( \frac{\partial f}{\partial y} \) or \( f_y \)

Example:

If \( z = x^3 y^2 + 5x \):
To find \( f_x \), treat \( y \) as a constant: \( f_x = 3x^2 y^2 + 5 \).
To find \( f_y \), treat \( x \) as a constant: \( f_y = 2x^3 y \).

Second Derivatives and Mixed Derivatives

Just like in normal calculus, you can differentiate again!
\( f_{xx} \): Differentiate \( f_x \) again with respect to \( x \).
\( f_{yy} \): Differentiate \( f_y \) again with respect to \( y \).
Mixed Derivatives (\( f_{xy} \) and \( f_{yx} \)): This means differentiating with respect to one variable and then the other.

The Mixed Derivative Theorem: For the functions you'll meet in this course, the order doesn't matter! \( f_{xy} = f_{yx} \). If you calculate both and they are different, it's a great way to spot a mistake!

Key Takeaway: Partial differentiation is "freezing" one variable to see how the height changes as you move in the direction of the other variable.


4. Stationary Points

On a 2-D curve, a stationary point is where the gradient is zero. On a 3-D surface, a stationary point is where the surface is flat in all directions. This happens when both first partial derivatives are zero:

\( f_x = 0 \) AND \( f_y = 0 \)

There are three main types of stationary points:

  1. Local Maximum: A peak.
  2. Local Minimum: A valley floor.
  3. Saddle Point: A point that looks like a maximum from one direction but a minimum from another (like the center of a pringle or a horse's saddle).

Key Takeaway: To find stationary points, solve the simultaneous equations \( f_x = 0 \) and \( f_y = 0 \).


5. Classifying Stationary Points (The Hessian Matrix)

In Stage 2, you need to be able to prove what kind of stationary point you've found. We use the Hessian Matrix, \( \mathbf{H} \):

\( \mathbf{H} = \begin{pmatrix} f_{xx} & f_{xy} \\ f_{yx} & f_{yy} \end{pmatrix} \)

The determinant of this matrix, \( |\mathbf{H}| = (f_{xx} \times f_{yy}) - (f_{xy})^2 \), tells us the nature of the point:

  • If \( |\mathbf{H}| > 0 \): It is a definite "turning point."
    • If \( f_{xx} > 0 \), it is a Local Minimum. (Think: positive = happy face/valley).
    • If \( f_{xx} < 0 \), it is a Local Maximum. (Think: negative = sad face/peak).
  • If \( |\mathbf{H}| < 0 \): It is a Saddle Point.
  • If \( |\mathbf{H}| = 0 \): The test is inconclusive (it could be anything!).

Mnemonic: "Hessian helps highlight high and low."

Key Takeaway: Calculate the second derivatives, find the determinant of the Hessian, and check the sign of \( f_{xx} \) to classify your point.


6. Tangent Planes

Just as a 2-D curve has a tangent line, a 3-D surface has a tangent plane at any given point \( (a, b) \).

Analogy: Imagine placing a flat piece of cardboard so it just touches a football at one specific point. That cardboard represents the tangent plane.

To find the equation of the tangent plane at point \( (a, b, f(a, b)) \), use this formula:

\( z = f(a, b) + (x - a)f_x(a, b) + (y - b)f_y(a, b) \)

Step-by-Step Process:
  1. Find the height \( z \) at the point by calculating \( f(a, b) \).
  2. Calculate the partial derivatives \( f_x \) and \( f_y \).
  3. Plug the coordinates \( (a, b) \) into your derivatives to get numerical gradients.
  4. Substitute everything into the tangent plane formula.

Key Takeaway: The tangent plane formula is just a 3-D version of the straight-line equation \( y - y_1 = m(x - x_1) \).


Summary Checklist

  • Can I distinguish between explicit (\( z=... \)) and implicit (\( g=... \)) surfaces?
  • Do I understand that a contour is a horizontal slice (\( z=c \))?
  • Can I perform partial differentiation while keeping one variable constant?
  • Do I remember that \( f_{xy} = f_{yx} \)?
  • Can I find stationary points by setting \( f_x = 0 \) and \( f_y = 0 \)?
  • Can I use the Hessian determinant to classify points as maxima, minima, or saddle points?
  • Can I construct the equation of a tangent plane at a specific point?

Keep practicing! Partial differentiation is very logical once you get used to "ignoring" one variable at a time. You've got this!