Prof. Vladimir Dobrushkin
Department of Mathematics

MTH243 (Calculus for Functions of Several Variables)
SAGE. Chapter 14:
Differentiating Functions of Several Variables

Vladimir A. Dobrushkin,Lippitt Hall 202C, 874-5095,

In this course we will use Sage computer algebra system (CAS), which is a free software. The Sage projects are created to help you learn new concepts. Sage is very useful in visualizing graphs and surfaces in three dimensions. Matlab (commercial software) is also available at engineeering labs. Its free version is called Octave. The university has a license for computer algebra system Mathematica, so it is free to use for its students. A student can also use free CASs: SymPy (based on Python), or Maxima.


Section 14.1. The Partial Derivative

Example 1. Plotting the Graph of the Function \( F (x, y) = x^2+y^2 \)
Plot3D[(x^2 + y^2), {x, -3, 3}, {y, -3, 3}, Axes -> True, PlotStyle -> Green]

Now we create a new graph: \( g (x, y) = x^2 + y^2 + 3 \)
Plot3D[(x^2 + y^2 + 3), {x, -3, 3}, {y, -3, 3}, Axes -> True]
Another graph of \( h(x, y) = 5 - x^2 - y^2 \)
Plot3D[(5 - x^2 - y^2), {x, -3, 3}, {y, -3, 3}, Axes -> True, PlotStyle -> Orange]
One more: \( k (x, y) = x^2 + (y - 1)^2 \)
Plot3D[(x^2 + (y - 1)^2), {x, -3, 3}, {y, -3, 3}, PlotStyle -> None]
Example 2. Plotting the Graph of the Function \( G(x,y)=e^{-(x^2+y^2)} \)
Plot3D[(E^-(x^2 + y^2)), {x, -5, 5}, {y, -5, 5}, PlotStyle -> Opacity[.8]]
Cross Sections and the Graph of a Function where x=2
Plot3D[{(x^2 + y^2), (4 + y^2)}, {x, -3, 3}, {y, -3, 3}]



Section 14.2. Computing Partial Derivatives


Section 14.3. Local Linearity and the Differential


Section 14.4. Gradients in the Plane


Section 14.5. Gradients in Space


Section 14.6. The Chain Rule

The fundamental thermodynamic relation may then be expressed in terms of the internal energy as:
\[ {\text d}U = T\,{\text d}S -p\,{\text d}V + \sum_{i=1}^s \mu_i {\text d}N_i , \]
which expresses the change in entropy \( {\text d}S \) of a system as a function of the intensive quantities temperature T, presure p, and ith chemical potential \( \mu_i \) and of the differentials of the intensive quantities energy U, volume V, and ith particle number \( N_i . \) The equation may be seen as a particular case of the chain rule. In other words
\begin{align*} \left( \frac{\partial U}{\partial S} \right)_{V, \{ N_i \}} &= T , \\ \left( \frac{\partial U}{\partial V} \right)_{S, \{ N_i \}} &= -p , \\ \left( \frac{\partial U}{\partial N_i} \right)_{S,V, \{ N_{j \ne i} \}} &= \mu_i . \end{align*}

These equations are known as "equations of state" with respect to the internal energy. (Note - the relation between pressure, volume, temperature, and particle number which is commonly called "the equation of state" is just one of many possible equations of state.) If we know all s+2 of the above equations of state, we may reconstitute the fundamental equation and recover all thermodynamic properties of the system.

The fundamental equation can be solved for any other differential and similar expressions can be found. For example, we may solve for  \( {\text d}S \) and find that

\[ \left( \frac{\partial S}{\partial V} \right)_{U, \{ N_i \}} = \frac{p}{T} . \]


Section 14.7. Second Order Partial Derivatives


Section 14.8. Differentiability