Six Pillars of Calculus (revisited)

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Calculus is built on six big ideas that I call the six pillars of calculus.

Close is good enough (limits)
Track the changes (derivatives)
What goes up has to stop before it comes down (max/min)
The whole is the sum of the parts (integration)
The whole change is the sum of the partial changes (fundamental theorem)
One variable at a time.

Most of multi-variable calculus is built on the last pillar, which we have already seen in the context of doing calculus with vector-valued functions. Partial derivatives combine the second and sixth pillars.

DNA forms a double helix, but the curvature of this helix depends on the temperature and on the salinity (concentration of salt). If we wanted to understand the curvature, we would do experiments by varying the conditions and measuring the curvature each time. (This can be done with gel electrophoresis. The tighter the DNA is coiled, the faster it moves through the gel.) If we did our experiments well, we wouldn't try changing both the temperature and the salinity. We would first hold the salinity fixed and change the temperature. Once we understood how temperature affects curvature, we would run a second set of experiments, holding the temperature fixed and varying the salinity. Combining the results, we would understand how both temperature and salinity affect curvature.

Mathematically, the curvature is a function $f(x,y)$, where $x$ is the temperature and $y$ is the salinity. Varying the temperature means comparing $f(x,y)$ to $f(x+h,y)$, and we can ask for the rate of change. Varying the salinity means comparing $f(x,y)$ to $f(x,y+h)$. By taking limits, we can compute two kinds of derivatives:

Definitions of partial derivatives:
The partial derivative of $f$ with respect to $x$ is $$\displaystyle{ \lim_{h \rightarrow 0} \frac{f(x+h,y)-f(x,y)}{h}}.$$
The partial derivative of $f$ with respect to $y$ is $$\displaystyle{ \lim_{h \rightarrow 0} \frac{f(x,y+h)-f(x,y)}{h}}$$

There are many notations for partial derivatives. If $z = f(x,y)$, then $$\displaystyle f_x(x,y) = f_x = \frac{\partial f}{\partial x}= \frac{\partial}{\partial x}f(x,y) = \frac{\partial z}{\partial x}= f_1 = D_1 f= D_x f$$ and $$\displaystyle f_y(x,y) = f_y = \frac{\partial f}{\partial y}= \frac{\partial}{\partial y}f(x,y) = \frac{\partial z}{\partial y}= f_2 = D_2 f= D_y f$$

The rough and precise definitions of limits of functions of two (or more) variables work the same way:

Rules for finding partial derivatives:

To find $f_x$, hold $y$ constant and differentiate with respect to $x$.
To find $f_y$, hold $x$ constant and differentiate with respect to $y$.

When computing $f_x$, we treat $y$ as a constant because it is a constant. After all, we are doing today's experiments at fixed salinity. This means that we can apply all of our familiar differentiation rules, pretending that the only variable is $x$.

Example: Compute $f_x$ and $f_y$ when $f(x,y) = \sin(x+y^2)$.

Solution: Since the derivative of $\sin(x + \hbox{ constant })$ with respect to $x$ is $\cos(x + \hbox{ constant })$, $$f_x = \cos(x+y^2).$$ Since the derivative of $\sin(\hbox{constant} + y^2)$ with respect to $y$ is $2y \cos(\hbox{constant }+y^2)$, $$f_y = 2y \cos(x+y^2).$$

In the following video, we review the Six Pillars and explain the ideas behind partial derivatives.

Integration by Parts

Integrals of Trig Functions

Trig Substitutions

Partial Fractions

Strategies of Integration

Improper Integrals

Modeling with Differential Equations

Euler's Method and Direction Fields

Separable Equations

Models of Growth

Linear Equations

Parametrized Curves

Calculus with Parametrized Curves

Polar Coordinates

Areas and Lengths of Polar Curves

Conic sections

Conic Sections in Polar Coordinates

Infinite Sequences

Infinite Series

Integral Test

Comparison Tests

Convergence of Series with Negative Terms

The Ratio and Root Tests

Strategies for testing Series

Power Series

Representing Functions as Power Series

Taylor and Maclaurin Series

Applications of Taylor Polynomials

Functions of 2 and 3 variables

Partial Derivatives

Differentiability and the Chain Rule

Multiple Integrals

Iterated Integrals over Rectangles

Double Integrals over General Regions

Double integrals in polar coordinates

Multiple integrals in physics

Integrals in Probability and Statistics

Change of Variables