
MTH243 (Calculus
for Functions of Several Variables)
MATLAB. Chapter 14:
Differentiating Functions of Several Variables
Vladimir A. Dobrushkin,Lippitt
Hall 202C, 8745095,dobrush@uri.edu
In this course we will use Mathematica computer algebra system (CAS), which is available in computer labs at URI. The
Mathematica projects are created to help you learn new concepts. Mathematica 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. A student can also use free CASs: SymPy (based on Python), Maxima, or Sage.

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
 