Welcome to Multivariable Calculus!
In standard A Level Maths, you’ve spent a lot of time looking at graphs in 2D—the classic \(y\) against \(x\). But the real world happens in 3D! In this chapter of the Extra Pure section, we are going to step into the third dimension. You’ll learn how to describe surfaces, find their "flat" points, and calculate exactly how steep a mountain is at any given point. Don't worry if this seems tricky at first; if you can differentiate a normal \(x^2\), you already have the skills to do this!
1. Surfaces in 3D: \(z = f(x, y)\)
In 2D, a function gives you a line. In 3D, a function like \(z = f(x, y)\) gives you a surface. Think of \(x\) and \(y\) as your coordinates on a floor, and \(z\) as the height of the ceiling above you at that point. If you do this for every point on the floor, you get a solid shape like a rolling hill or a bowl.
Slicing it up: Contours and Sections
Visualising 3D shapes on 2D paper is hard! To help, we use two main tricks:
- Contours: These are lines where the height \(z\) is constant. Analogy: Think of a standard hiking map. The circles represent different heights. If you walk along a contour line, you stay at the exact same altitude.
- Sections: This is like taking a giant knife and slicing through the surface.
- A section of the form \(z = f(a, y)\) is a slice parallel to the \(y\)-axis (where \(x\) is fixed at some value \(a\)).
- A section of the form \(z = f(x, b)\) is a slice parallel to the \(x\)-axis (where \(y\) is fixed at some value \(b\)).
Did you know? Meteorologists use multivariable calculus to model weather systems. The "surfaces" they study represent air pressure or temperature across a map!
Key Takeaway: A 3D surface can be understood by looking at its "shadows" (contours) or by "slicing" it (sections) to see what the 2D curve looks like at that specific spot.
2. Partial Differentiation
If we have a surface, we want to know how steep it is. But there's a catch: it might be steep if you walk North, but flat if you walk East! To solve this, we use Partial Derivatives.
How to do it:
We use a curly 'd' symbol: \(\partial\).
- \(\frac{\partial z}{\partial x}\) means "Differentiate with respect to \(x\), and pretend \(y\) is just a constant number (like 5)."
- \(\frac{\partial z}{\partial y}\) means "Differentiate with respect to \(y\), and pretend \(x\) is just a constant number."
Example: If \(z = x^2y + 3y^3\)
To find \(\frac{\partial z}{\partial x}\): Treat \(y\) as a constant. The derivative of \(x^2y\) is \(2xy\). The derivative of \(3y^3\) is \(0\) (because there is no \(x\) there).
So, \(\frac{\partial z}{\partial x} = 2xy\).
Common Mistake to Avoid: Don't try to use the product rule on \(x^2y\) when finding \(\frac{\partial z}{\partial x}\) unless \(y\) is actually a function of \(x\). Here, \(y\) is independent, so it’s just a "coefficient" like a number.
Quick Review:
1. Look at the variable you are differentiating by.
2. Every other variable is treated like a static number.
3. Differentiate as usual!
3. Stationary Points: Summits, Valleys, and Saddles
Just like in 2D, a stationary point is where the surface is flat. For this to happen, the surface must be flat in both the \(x\) and \(y\) directions at the same time.
The Condition:
To find stationary points, you must solve these two equations simultaneously:
$$\frac{\partial z}{\partial x} = 0 \text{ and } \frac{\partial z}{\partial y} = 0$$
Types of Stationary Points:
- Local Maximum: The top of a hill.
- Local Minimum: The bottom of a bowl.
- Saddle Point: This is a special point that looks like a maximum from one direction but a minimum from another.
Analogy: Think of a Pringle chip or a horse saddle. If you sit on the saddle, it goes up in front and behind you (minimum), but it slopes down to your left and right (maximum).
Note: In this syllabus, if you are asked to find the *nature* of the point (whether it's a max, min, or saddle), the method will be given to you in the question. You just need to know how to find the coordinates!
Key Takeaway: Stationary points occur where both partial derivatives are zero.
4. The Gradient Vector: \(\text{grad } g\)
Sometimes surfaces are defined implicitly, like \(g(x, y, z) = c\). The gradient vector (often written as grad \(g\) or \(\nabla g\)) is a vector that points in the direction of the steepest increase at a specific point.
The Formula:
The gradient vector is simply a column vector of the partial derivatives:
$$\text{grad } g = \begin{pmatrix} \frac{\partial g}{\partial x} \\ \frac{\partial g}{\partial y} \\ \frac{\partial g}{\partial z} \end{pmatrix}$$
If your surface is given as \(z = f(x, y)\), you can rewrite it as \(z - f(x, y) = 0\). The formula for the gradient vector in this specific case is:
$$\text{grad } g = \begin{pmatrix} \frac{\partial f}{\partial x} \\ \frac{\partial f}{\partial y} \\ -1 \end{pmatrix}$$
Important Property: The gradient vector at a point is always perpendicular (normal) to the surface at that point. This is the "Golden Rule" for the next section!
5. Tangent Planes and Normal Lines
Imagine placing a flat piece of cardboard so it just touches a curved surface at one point. That's the Tangent Plane. Now imagine a flagpole sticking straight out of the surface at 90 degrees. That's the Normal Line.
The Tangent Plane:
Since the gradient vector \(\mathbf{n} = \text{grad } g\) is normal to the surface, it is also the normal vector of the tangent plane.
Using the vector equation of a plane \(\mathbf{r} \cdot \mathbf{n} = \mathbf{a} \cdot \mathbf{n}\), where \(\mathbf{a}\) is your point \((x_0, y_0, z_0)\):
$$\frac{\partial g}{\partial x}(x - x_0) + \frac{\partial g}{\partial y}(y - y_0) + \frac{\partial g}{\partial z}(z - z_0) = 0$$
The Normal Line:
The normal line passes through point \(\mathbf{a}\) and goes in the direction of the gradient vector.
The vector equation is: \(\mathbf{r} = \mathbf{a} + \lambda(\text{grad } g)\).
Memory Aid:
The Gradient Vector is the "key" to both!
- For the Plane: The gradient vector is the "direction it's facing" (the normal).
- For the Line: The gradient vector is the "direction it's traveling."
Key Takeaway: Calculate the partial derivatives, put them in a column vector, and use your Core Pure vector knowledge to build the line or plane equations.
Final Summary Review
- Surfaces: Defined by \(z = f(x, y)\) or \(g(x, y, z) = c\).
- Partial Derivatives: Differentiate one variable, hold the others constant.
- Stationary Points: Solve \(\frac{\partial z}{\partial x} = 0\) and \(\frac{\partial z}{\partial y} = 0\).
- Grad: A vector of partial derivatives, perpendicular to the surface.
- Normal/Tangent: Use the Grad vector as your normal vector \(\mathbf{n}\) or direction vector \(\mathbf{d}\).
You've got this! Multivariable calculus is just taking what you know about 2D and applying it twice. Keep your variables straight, and the marks will follow!