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Multivariable Calculus

Calculus in 3D: Functions, Surfaces, and Fields.

Single-variable calculus deals with lines and curves on a flat piece of paper (the x-y plane). But the real world is three-dimensional. Multivariable Calculus extends the rules of differentiation and integration to functions with more than one input, allowing us to analyze surfaces, hills, valleys, and fluid flows.

1. Functions of Several Variables

In standard calculus, we write y = f(x). This gives us a 2D line.
In multivariable calculus, we write z = f(x, y). This gives us a height (z) for every coordinate pair (x, y).

[Image of 3D paraboloid surface plot]

When you graph this, you don't get a curve; you get a surface. Think of a topographic map or a mountainous landscape.

2. Partial Derivatives

If you are standing on the side of a mountain, the slope depends on which direction you look. You might face uphill (steep slope) or sideways (zero slope).

A Partial Derivative measures the rate of change with respect to one variable while holding the other variables constant.

[Image of partial derivative tangent plane diagram]
Notation: We use the curly "∂" (del).
∂f/∂x = Slope in x-direction (treat y as constant)
∂f/∂y = Slope in y-direction (treat x as constant)

Example

Let f(x, y) = x²y + 3y³.

  • Find ∂f/∂x: Treat y like a number (like 5).
    Derivative of x²y is 2xy. Derivative of 3y³ is 0 (constant).
    Result: 2xy
  • Find ∂f/∂y: Treat x like a number.
    Derivative of x²y is x²(1). Derivative of 3y³ is 9y².
    Result: x² + 9y²

3. The Gradient Vector (∇f)

If you combine all the partial derivatives into a vector, you get the Gradient, denoted by the symbol ∇ (nabla).

∇f = < ∂f/∂x, ∂f/∂y >

The gradient is incredibly useful because it always points in the direction of steepest ascent. If you are on a hill and want to climb up as fast as possible, just follow the gradient vector.

4. Multiple Integrals

Just as a single integral finds the area under a curve, a Double Integral finds the volume under a surface.

[Image of double integral volume under surface diagram]
Volume = ∬ f(x, y) dA

This involves integrating with respect to x, and then integrating the result with respect to y (or vice-versa). We can extend this to Triple Integrals to calculate properties of 3D solids, like their mass or center of gravity.

5. Applications

Multivariable calculus is the language of the physical universe:

  • Electromagnetism: Maxwell's equations describe electric and magnetic fields using multivariable vector calculus.
  • Fluid Dynamics: Modeling how air flows over an airplane wing.
  • Machine Learning: Training a neural network involves "Gradient Descent"—finding the minimum error in a massive multivariable landscape.