Having studied math for more than fifteen years, one fact I have learned is that there is always more math to learn. In the case of aerospace engineering (and all other engineering disciplines), there is some really useful and interesting math that goes beyond single-variable calculus and ordinary differential equations. In this post, we will explore this new, difficult, and interesting math.

In the previous blog post about the math of aerospace engineering, we discussed a branch of mathematics called calculus. What we really discussed, however, was single-variable calculus, or the calculus of functions that have only one variable. In reality, there are numerous functions that have more than one input. For example, the temperature on Earth varies based on latitude, longitude, and altitude. Therefore, it is useful to extend our notions of integration and differentiation to the world of multivariable functions.

If we have a function that takes in two variables, x and y, and returns a scalar, f(x,y), we can define two different derivatives. First, we can determine the rate of change of f with respect to x, denoted by ∂f/∂x. Here, we treat x as a variable and y as a constant as we differentiate. Second, we can determine the rate of change of f with respect to y, denoted by ∂f/∂y. In this case, we treat y as a variable and x as a constant as we differentiate. We can extend this notion of partial differentiation to functions of any number of variables, where when we take ∂f/∂{some variable}, we treat that variable as a quantity subject to change and all other variables as constants.

Meanwhile, we can define integrals for functions of any number of variables. While integrals for functions of three or more variables do not have easy geometric interpretations, we can look at an integral of a function of two variables, f(x,y), as finding the volume under the surface that makes up the graph of f(x,y) in three-dimensional space. When integrating multivariable functions, it is important to recognize that we need as many integral signs as we have variables. For example, the volume under the surface of a two-variable function can be written as:

∫∫_{G} f(x,y) dxdy.

The integral of a three-variable function can be written as:

∫∫∫_{G} f(x,y,z) dxdydz.

For these higher-dimensional integrals, we are effectively taking infinitely tiny chunks of a domain that we call “G” (where these chunks are multi-dimensional), then multiplying these chunks by the value of the function over those chunks, and finally summing up all of these products of domain chunks and function values.

Now we have enough machinery to revisit differential equations. Our discussion in the last blog post about math mostly concerned ordinary differential equations, or differential equations where we are solving for a function that has only one variable. For example, dy/dx = y was an ordinary differential equation because the function we were solving for, y(x), had only one variable, namely x.

When dealing with multivariable functions, we can have differential equations where the function we are solving for has more than one variable. These are called partial differential equations, and they show up all over advanced math, physics, and engineering. Three of the most famous partial differential equations are the Heat Equation, Wave Equation, and Laplace’s Equation.

The Heat Equation can tell us how the temperature, denoted by u(x,y,z,t), on a three dimensional body can evolve over time, and frequently shows up as:

∂u/∂t = k ∆u, where k is a constant and ∆u = ∂^{2}u/∂x^{2} + ∂^{2}u/∂y^{2} + ∂^{2}u/∂z^{2}. (∆u is also called the Laplacian of u.)

∂^{2}u/∂x^{2}, ∂^{2}u/∂y^{2}, and ∂^{2}u/∂z^{2} represent the second-derivatives of u with respect to our spatial variables. In other words, they are partial derivatives of partial derivatives.

The Wave Equation can tell us how the height of a wave or vibrating chord can change over time and space, and frequently shows up as:

∂^{2}u/∂t^{2} = c^{2} ∆u, where c^{2} is a positive constant.

Lastly, Laplace’s Equation is compactly written as:

∆u = 0.

In two spatial dimensions, ∆u = ∂^{2}u/∂x^{2} + ∂^{2}u/∂y^{2}, and in three spatial dimensions, ∆u = ∂^{2}u/∂x^{2} + ∂^{2}u/∂y^{2} + ∂^{2}u/∂z^{2}.

Finding solutions to partial differential equations is a beast of its own, and I hope to explore that topic in a future blog post. For now, we have just about run out of time. We have, however, successfully given some important definitions and terminology that are essential in advanced engineering mathematics. I wish all my readers the best this week, and I look forward to returning with a new post next week.

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