MULTIVARIABLE CALCULUS 01
Functions of Several Variables
A function of several variables takes more than one input and produces an output.
The main idea
A function of one variable may be written as
\[ y=f(x). \]There is one input, \(x\), and one output, \(y\).
A function of two variables may be written as
Now the output \(z\) depends on two inputs, \(x\) and \(y\).
1. A simple example
Consider
\[ f(x,y)=x+y. \]If
\[ x=2, \qquad y=3, \]then
\[ f(2,3)=2+3=5. \]So the ordered pair
\[ (2,3) \]is the input and
\[ 5 \]is the output.
2. Another example
Let
\[ f(x,y)=x^2+y^2. \]Then
\[ f(1,2) = 1^2+2^2 = 5. \]And
\[ f(3,4) = 3^2+4^2 = 25. \]3. Input as a point
For a function
\[ f(x,y), \]the input is not a single number. It is a point in the plane:
\[ (x,y). \]So a function of two variables can be viewed as assigning a value to each point in a region of the \(xy\)-plane.
4. Geometric interpretation
For
\[ z=f(x,y), \]each point
\[ (x,y) \]in the input plane is assigned a height
\[ z. \]The graph is therefore a surface in three-dimensional space.
5. Example surface
Consider
\[ z=x^2+y^2. \]For each point \((x,y)\), the height is
\[ x^2+y^2. \]The smallest value occurs at
\[ (0,0), \]where
\[ z=0. \]As we move away from the origin, \(z\) increases.
6. Surface diagram
7. Domain
The domain of a multivariable function is the set of all allowed input points.
For example,
\[ f(x,y)=\sqrt{4-x^2-y^2}. \]For real outputs, we need
\[ 4-x^2-y^2\ge0. \]Therefore,
\[ x^2+y^2\le4. \]So the domain is the disk of radius \(2\) centred at the origin.
8. Range
The range is the set of possible output values.
For
\[ f(x,y)=x^2+y^2, \]we always have
\[ x^2+y^2\ge0. \]So the range is
\[ [0,\infty). \]9. More than two variables
A function can depend on three or more variables.
For example,
\[ T(x,y,z) \]could represent temperature at a point in three-dimensional space.
Or
\[ P(S,I,\beta,\gamma) \]could represent a quantity depending on several model variables and parameters.
10. Why graphs become difficult
A function
\[ z=f(x,y) \]can be represented by a surface in three dimensions.
But a function of three inputs,
\[ w=f(x,y,z), \]would require four dimensions to graph directly.
So multivariable calculus often uses other visual tools, such as level curves and vector fields.
11. Level curves
A level curve is obtained by fixing the output at a constant value:
\[ f(x,y)=c. \]For
\[ f(x,y)=x^2+y^2, \]a level curve satisfies
\[ x^2+y^2=c. \]These are circles.
12. Why level curves are useful
Instead of trying to see the full three-dimensional surface, we can look at curves of equal height.
This is similar to contour lines on a map.
13. Multivariable change
For a function
\[ z=f(x,y), \]the output can change because \(x\) changes, because \(y\) changes, or because both change.
This creates a new question:
How does \(z\) change when only one input changes and the others are held fixed?
14. This leads to partial derivatives
To measure change with respect to \(x\) while holding \(y\) fixed, we use
\[ \frac{\partial f}{\partial x}. \]To measure change with respect to \(y\) while holding \(x\) fixed, we use
\[ \frac{\partial f}{\partial y}. \]15. A modelling example
Suppose
\[ C(q,p)=pq+q^2. \]The quantity \(C\) depends on two inputs:
\[ q \qquad\text{and}\qquad p. \]Changing \(q\) while keeping \(p\) fixed may affect \(C\) differently from changing \(p\) while keeping \(q\) fixed.
Multivariable calculus gives us a systematic way to distinguish these effects.
16. Scalar-valued functions
A function such as
\[ f(x,y)=x^2+y^2 \]takes several inputs but produces one numerical output.
This is called a scalar-valued function.
The input is a vector or point, while the output is a scalar.
17. Vector-valued functions
A function can also return several outputs.
For example,
\[ \mathbf{F}(x,y) = \begin{pmatrix} -y\\ x \end{pmatrix}. \]This assigns a vector to every point \((x,y)\).
Such functions lead naturally to vector fields, divergence and curl.
18. The central idea
becomes
\[ \boxed{ (x,y) \longrightarrow f(x,y) } \]The main change is simple: instead of one independent variable, the function has several.
But this creates richer ideas of direction, partial change and geometry.