Functions of Several Variables

In real analysis, the concept of functions of several variables is an extension of single-variable functions into several dimensions. These functions are essential not only for understanding advanced mathematics but also for understanding many real-world applications in physics, engineering, economics, and beyond.

Basics of functions with multiple variables

A function with multiple variables takes inputs from multiple dimensions and maps them to an output, which is typically in one dimension. Mathematically, such a function can be written as:

    f: ℝⁿ → ℝ

Here, ℝⁿ represents the n-dimensional space of the input variables, and ℝ is the real number line where the output resides.

For example, consider a function f(x, y) that takes two real numbers, x and y, and maps them to a real number. In real-world applications, these can represent any measurable quantity such as temperature, pressure, etc.

Graphical representation

Unlike a single-variable function, which can be viewed in two dimensions, a function of two variables requires three dimensions to be viewed. The graph of such a function f(x, y) is a surface in three-dimensional space.

Above is a simple 2D representation showing the surface created by the function f(x, y). Here, changing x and y will result in different z values, creating the surface.

Partial derivative

An important concept in differentiating functions of several variables is the idea of the partial derivative. The partial derivative of a function with respect to one variable is essentially the derivative of the function, assuming all other variables are constant.

For a function f(x, y), the partial derivative of f with respect to x is represented by ∂f/∂x. Similarly, the partial derivative with respect to y is ∂f/∂y.

    ∂f/∂x = lim (h → 0) [(f(x+h, y) - f(x, y)) / h]

The above expression gives the rate of change of f in x direction, keeping y constant.

Example of partial derivative

Let's consider a simple function:

    f(x, y) = x²y + y³

Find the partial derivative of f with respect to x:

    ∂f/∂x = ∂/∂x (x²y + y³) = 2xy

Here, the term y³ vanishes because it is constant with respect to x.

Similarly, finding the partial derivative of f with respect to y:

    ∂f/∂y = ∂/∂y (x²y + y³) = x² + 3y²

Higher-order derivatives

In many applications, it may be necessary to further calculate derivatives, leading to higher-order derivatives. These are derivatives of partial derivatives. For example, the second derivative of f with respect to x can be written as:

    ∂f²/∂x²

Similarly, we can have mixed partial derivatives such as:

    ∂²f/(∂x∂y)

Mixed partial derivatives can be computed in any order, and in many cases, they are equal, a fact known as Clair's theorem.

Example problem of higher-order derivatives

Continuing our previous example f(x, y) = x²y + y³, let's calculate the mixed second-order partial derivative:

1. First, find ∂f/∂x = 2xy.
2. Now differentiate ∂²f/(∂x∂y) with respect to y:

    ∂²f/(∂x∂y) = ∂/∂y (2xy) = 2x

Similarly, applying the series of differentiation, we deal with ∂²f/(∂y∂x):

    ∂f/∂y = x² + 3y²

    ∂²f/(∂y∂x) = ∂/∂x(x² + 3y²) = 2x

As expected, ∂²f/(∂x∂y) = ∂²f/(∂y∂x) = 2x, showing that the mixed derivatives are equal.

Gradient and interpretation

The gradient of a function of several variables is a vector that points in the direction of the greatest rate of increase of the function. For the function f(x, y, ...), the gradient is:

    ∇f = (∂f/∂x, ∂f/∂y, ...)

The gradient thus forms a vector whose components are the partial derivatives of f. This can be interpreted as the slope or inclination in the corresponding coordinate directions.

Practical example

Let f(x, y) = 3x² + 4xy + 2y², then:

    ∇f = (∂f/∂x, ∂f/∂y)
    ∂f/∂x = 6x + 4y
    ∂f /∂y = 4x + 4y

    ∇f = (6x + 4y, 4x + 4y)

The gradient ∇f at any point (x, y) gives the direction and rate of steepest ascent in a two-dimensional plane.

Understanding the Hessian matrix

The Hessian matrix is a square matrix of second-order mixed partial derivatives of a scalar-valued function. For a function f(x, y), the Hessian can be represented as:

    h(f) = | ∂²f/∂x² ∂²f/(∂x∂y) |
           | ∂²f/(∂y∂x) ∂²f/∂y² |

The Hessian is often used to determine the concavity or convexity of functions in optimization problems.

Application example of Hessian

Let's revisit our previous example f(x, y) = x²y + y³:

    h(f) = | 2y 2x |
           | 2x 6y² |

Calculating the determinant and checking whether the matrix is positive definite helps in determining the nature of the function at critical points.

Taylor series expansion

Just as a single-variable function can be approximated using a Taylor series, a function with multiple variables can also be expanded in a similar way. The Taylor series for the function f(x, y) around the point (a, b) is given by:

    f(x, y) ≈ f(a, b) + (xa)fₓ(a, b) + (yb)fᵧ(a, b) 
    + 1/2 [(xa)² fₓₓ(a, b) + 2(xa)(yb)fₓᵧ(a, b) + (yb)²fᵧᵧ(a, b)]

This expansion involves terms of increasing complexity and provides a systematic way of approximating functions near particular points.

Conclusion

Functions of several variables are a fundamental part of modern mathematics, with applications spanning many scientific disciplines. Their differentiation concepts, such as partial derivatives, gradients, and Hessian matrices, are important tools for mathematical modeling and analysis.

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