Understanding Multivariable Calculus: Problems, Solutions, and Tips

Reviewing key concepts from basic calculus; a brief overview of what is to be learned; applying distance and midpoint formulas to three-dimensional objects; observing some of the new curiosities unique to functions of more than one variable.

What makes a function "multivariable"; beginning with definitions; how these new functions behave as one applies familiar concepts of minimum and maximum values; using graphics and other tools to observe their interactions with the xy-plane.

Applying fundamental definitions of calculus to multivariable functions, starting with their limits; how these limits become complicated as one approaches them; using this to derive the definition of a versatile new tool, the partial derivative.

Discovering the new dimensions of second partial derivatives; differentiating either twice with respect to x or y, or with respect once each to x and y; considering Laplace's equation to see what makes a function "harmonic".

Combining the differential and chain rule from elementary calculus; how to generalize them to functions of more than one variable; how the total differential can be used to approximate ?z over small intervals without calculating the exact values.

The ability to find extreme values for optimization is one of the most powerful consequences of differentiation; defining the Extreme Value theorem for multivariable functions; using it to identify relative extrema using a "second partials test".