Welcome to 3D: Surfaces and Partial Differentiation
In standard A Level Maths, you spent a lot of time looking at curves on a flat 2D grid. In Further Mathematics, we take the leap into the third dimension! Instead of just finding the slope of a line, we are going to look at surfaces—think of hills, valleys, and even the shape of a Pringles chip.
Don’t worry if this seems tricky at first. We are just taking the differentiation rules you already know and applying them to one variable at a time. It’s like looking at a 3D object from different sides to understand its shape. Let's dive in!
1. What is a 3-D Surface?
A surface is defined by a function of two variables, usually written as \(z = f(x, y)\). This means the "height" (\(z\)) depends on where you are on the floor (\(x\) and \(y\)).
Explicit form: \(z = x^2 + 3xy\). Here, \(z\) is clearly the subject.
Implicit form: \(x^2 + y^2 + z^2 = 25\). This is a sphere where all variables are mixed together.
Analogy: Imagine a mountain range. If you give me your GPS coordinates (\(x, y\)), the function \(f(x, y)\) tells me exactly how high up you are standing (\(z\)).
Quick Review: The Basics
To succeed here, you just need to be comfortable with Power Rule differentiation from your pure core studies. If you can differentiate \(y = x^n\), you can do this!
Key Takeaway: An equation with \(x, y,\) and \(z\) represents a surface in 3D space, just like an equation with \(x\) and \(y\) represents a curve in 2D space.
2. Sections and Contours: Slicing the Surface
Visualising a 3D surface on a 2D piece of paper is hard. Mathematicians use two main tricks to "squash" the 3D surface into something easier to see: Sections and Contours.
Sections (Vertical Slices)
A section is what you get when you slice a surface with a vertical plane. You do this by keeping either \(x\) or \(y\) constant.
- If you set \(x = a\) (a constant), you get a section parallel to the \(yz\)-plane.
- If you set \(y = b\) (a constant), you get a section parallel to the \(xz\)-plane.
Example: If \(z = x^2 + y^2\), and we look at the section where \(x = 2\), the equation becomes \(z = 4 + y^2\). This is just a 2D parabola!
Contours (Horizontal Slices)
A contour is what you get when you slice the surface horizontally at a fixed height. You do this by setting \(z = c\) (a constant).
Real-world Example: Think of a topographical map used for hiking. The lines on the map are contours. If you walk along a contour line, you stay at the exact same altitude.
Key Takeaway: Sections help us see the side profile of a surface, while contours show us the view from directly above.
3. Partial Differentiation: One Variable at a Time
Partial differentiation sounds scary, but it’s actually a "cheat code." When we find a partial derivative, we differentiate with respect to one variable and pretend the other one is just a regular number (a constant).
The Notation
We use a "curly d" (\(\partial\)) to show we are doing partial differentiation:
\(\frac{\partial z}{\partial x}\) (pronounced "partial dz by dx") or simply \(f_x\).
How to do it (Step-by-Step)
Let's find the partial derivatives of \(z = x^2y + 5x + y^3\):
Step 1: Find \(f_x\) (Differentiate \(x\), treat \(y\) as a constant).
The \(x^2y\) becomes \(2xy\).
The \(5x\) becomes \(5\).
The \(y^3\) has no \(x\) in it, so it's treated like a constant number (like "7") and disappears (becomes 0).
Answer: \(f_x = 2xy + 5\).
Step 2: Find \(f_y\) (Differentiate \(y\), treat \(x\) as a constant).
The \(x^2y\) becomes \(x^2\) (because the derivative of \(y\) is 1).
The \(5x\) has no \(y\) in it, so it disappears.
The \(y^3\) becomes \(3y^2\).
Answer: \(f_y = x^2 + 3y^2\).
Second Derivatives and the Mixed Derivative Theorem
You can differentiate again to find second-order derivatives, like \(f_{xx}\) or \(f_{yy}\).
There are also mixed derivatives, like \(f_{xy}\) (differentiate by \(x\), then by \(y\)).
Did you know? For the functions you'll see in this course, \(f_{xy} = f_{yx}\). It doesn't matter which order you differentiate in; you'll get the same result! This is known as the Mixed Derivative Theorem.
Key Takeaway: To find a partial derivative, focus only on the variable you are asked for and treat the other like a boring constant number.
4. Finding Stationary Points
In 2D, a stationary point is where the slope is zero (\(\frac{dy}{dx} = 0\)). In 3D, a stationary point is where the surface is perfectly flat in every direction.
The Conditions
For a point to be stationary, both partial derivatives must be zero at the same time:
\(f_x = 0\) and \(f_y = 0\).
Types of Stationary Points
- Local Maximum: The peak of a hill.
- Local Minimum: The bottom of a bowl.
- Saddle Point: A very cool shape! It looks like a mountain pass. It's a maximum if you move in one direction (e.g., \(x\)) but a minimum if you move in the other (e.g., \(y\)). Think of a horse saddle or a Pringles chip.
Common Mistake: Students often find where \(f_x = 0\) and stop. Remember, you must solve for \(x\) and \(y\) using both equations simultaneously (often using simultaneous equations).
Quick Review Box:
1. Differentiate to find \(f_x\) and \(f_y\).
2. Set \(f_x = 0\) and \(f_y = 0\).
3. Solve for \(x\) and \(y\).
4. Plug \(x\) and \(y\) back into the original \(z\) equation to find the height.
Key Takeaway: Stationary points occur where the surface is temporarily flat. You find them by setting both partial derivatives to zero.
Summary Checklist
Before you move on, make sure you can:
• Distinguish between sections (vertical slices) and contours (horizontal slices).
• Calculate first and second partial derivatives using power rules.
• Remember that \(f_{xy} = f_{yx}\).
• Set up and solve \(f_x = 0\) and \(f_y = 0\) to find stationary points.
You're doing great! Surfaces and partial derivatives are a fundamental tool used by engineers and data scientists every day. Keep practicing, and the "3D vision" will become second nature!