Calculus in Materials Science and Engineering (MATSE)

Although calculus seems irrelevant and unimportant while you are sitting in class, it truly does prove relevant as I have come to find through a sit down with Dr. R. Allen Kimel.

            In my meeting with Dr. R. Allen Kimel I saw an example of calculus in the form of Fick’s Laws of Diffusion.  This is very important in MATSE because this process makes materials stronger in terms of stiffness or hardness.  The formula will help us determine how far into a material a substance diffuses and what the concentration is where in the material.  The reason this is calculus is because the formula deals with derivatives. 
J= - D dC/dx where D is a diffusion constant, J is the flux of the material, C is concentration, and x is the position.  This specific formula is used when the concentration is steady.  However if the concentration is not steady then we must use the formula dC/dt = D d2C/dx2 where all the other variables are the same and t is time.  A practical application of this process is carbonizing.  In order to make metals harder, you can introduce carbon compounds to them under the right conditions and the carbon will diffuse into the metal.  So in order to find how far in the carbon will diffuse at what concentration or what the concentration is at a certain distance we can use these formulas.

            This is a close up of a gear that is hardened by carbonization and the outer edge of the teeth has a high density of carbon and it lessens as position increases to the inside of the each tooth.  Learning about this example has changed my thinking of calculus a little.  I always knew there were practical uses for calculus, but I knew very few examples of it.  This law shows that calculus is used all the time in MATSE because hardening materials is very important and the figuring out the amount before you actually change the material means that you can predict what properties the new material will have.