Function
of Two variable: Let u
be a symbol which has a definite value for every pair of values of x and y,
then u is called a function of two independent variable x and y and is written
as
u=f(x,y)
Function of Two Variables – Graphical Representation
A function of two variables is written as:
z = f(x, y)
1. General 3D Graph
z
|
| •
| •
| •
| •
|•____________ y
/
/
x
Explanation: This shows a surface in 3D space where z depends on x and y.
2. Plane Surface (z = x + y)
z
|
| /
| /
| /
|/________ y
/
/
x
Application: Cost, temperature variation.
3. Paraboloid (z = x² + y²)
z
|
__|__
/ | \
/ | \
/_______|_______\ y
|
x
Application: Heat distribution, potential energy.
4. Saddle Surface (z = xy)
z
|
___/ \___
/ \
----/-------------\---- y
\ /
\___ ___/
x
Application: Profit–loss analysis.
5. Contour Diagram (x² + y² = c)
y
|
○ ○ ○
○ ○
○ ○ ○
○ ○
○ ○
|
x
Application: Topographic maps, weather maps.
Exam Note:
The graph of a function of two variables is a surface in three-dimensional space.
Function of Three Variables
A function of three variables is a function that depends on three independent variables.
Mathematical Form:
w = f(x, y, z)
Examples of Functions of Three Variables
- Linear: f(x, y, z) = x + y + z
- Quadratic: f(x, y, z) = x² + y² + z²
- Product: f(x, y, z) = xyz
- Trigonometric: f(x, y, z) = sin x + cos y + tan z
- Rational: f(x, y, z) = (x + y) / z, z ≠ 0
Graphical Representation
A function of three variables cannot be drawn directly because it requires four dimensions.
Hence, it is represented using level surfaces or cross-sections.
Level Surface Example
For: x² + y² + z² = c (Sphere)
z
|
___|___
.-' | '-.
.' | '.
| | |
'. | .'
'-._____|_____.-'
|
x
/
y
Explanation: Each surface represents points where the function has the same value.
Cross-Section Concept
Fix one variable (e.g., z = k) and draw the 3D surface in x–y plane.
z = k
|
____|____
/ | \
| | |
\____|____/
|
x
/
y
Real-Life Applications
- Temperature in a room: T = f(x, y, z)
- Pressure in fluids
- Electric and gravitational potential
- Air pollution concentration
- Medical imaging (CT, MRI)
Exam Note:
A function of three variables is represented graphically using level surfaces or cross-sections.
Assignment
Functions of Two and Three Variables
Functions of Two and Three Variables
Section 1: Functions of Two Variables
Definition
A function of two variables depends on two independent variables x and y and produces a single dependent variable z:
z = f(x, y)
Examples
- f(x, y) = x + y
- f(x, y) = x² + y²
- f(x, y) = xy
- f(x, y) = sin x + cos y
- f(x, y) = x / y, y ≠ 0
Graphical Representation
- Graph is a surface in 3D space.
- Plane: z = x + y
- Paraboloid: z = x² + y²
- Saddle: z = xy
Level Curves
A level curve is given by f(x,y) = c, e.g., x² + y² = 1.
Applications
- Temperature distribution
- Population density
- Cost analysis
- Pressure on surfaces
- Topography
Section 2: Functions of Three Variables
Definition
A function of three variables depends on x, y, z and produces a single output w:
w = f(x, y, z)
Examples
- f(x,y,z) = x + y + z
- f(x,y,z) = xyz
- f(x,y,z) = x² + y² + z²
- f(x,y,z) = sin x + cos y + tan z
- f(x,y,z) = (x+y)/z, z ≠ 0
Graphical Representation
- Cannot be plotted directly in 3D (requires 4D).
- Use level surfaces: f(x,y,z) = c
- Use cross-sections: fix one variable and plot remaining 2D function
Applications
- Temperature in a room
- Pressure in fluids
- Air pollution modeling
- Electric/gravitational fields
- Medical imaging (CT/MRI)
Section 3: Questions and Answers
Functions of Two Variables
Q1: Define a function of two variables.
A1: A function of two variables depends on x and y and produces a single output z.
Q2: Write the general form of a function of two variables.
A2: z = f(x, y)
Q3: How many dimensions are needed to graph it?
A3: Three dimensions (x, y, z)
Q4: What is a level curve? Give an example.
A4: A curve where f(x,y) = c, e.g., x² + y² = 1
Q5: Give any two examples.
A5: f(x,y)=x+y, f(x,y)=x²+y²
Q6: If f(x,y)=x²+y², find f(2,3).
A6: f(2,3)=2²+3²=13
Q7: Find f(-2,4) for f(x,y)=xy.
A7: f(-2,4)=(-2)(4)=-8
Q8: Determine domain of f(x,y)=x/y.
A8: All real x,y with y≠0
Functions of Three Variables
Q1: Define a function of three variables.
A1: Depends on x, y, z and produces a single output w.
Q2: Write the general form.
A2: w = f(x, y, z)
Q3: Why cannot it be plotted directly?
A3: Requires 4D visualization; use level surfaces or cross-sections.
Q4: Give three examples.
A4: f(x,y,z) = x+y+z, f(x,y,z) = xyz, f(x,y,z)=x²+y²+z²
Q5: What is a level surface? Give example.
A5: Surface where f(x,y,z)=c, e.g., x²+y²+z²=1 (sphere)
Q6: If f(x,y,z)=x+y+z, find f(1,2,3).
A6: 6
Q7: If f(x,y,z)=xyz, find f(2,-1,3).
A7: -6
Q8: Determine domain of f(x,y,z)=(x+y)/z.
A8: All real x,y,z with z≠0
Section 4: Diagrams
General Graph of Function of Two Variables
z
|
| •
| •
| •
| •
|•____________ y
/
/
x
Plane Surface (z = x + y)
z
|
| /
| /
| /
|/________ y
/
/
x
Paraboloid (z = x² + y²)
z
|
__|__
/ | \
/ | \
/_______|_______\ y
|
x
Saddle Surface (z = xy)
z
|
___/ \___
/ \
/-------------\---- y
\ /
\___ ___/
x
Level Curve (x² + y² = c)
y
|
○ ○ ○
○ ○
○ ○ ○
○ ○
○ ○
|
x
Level Surface for Function of Three Variables (x² + y² + z² = c)
z
|
___|___
.-' | '-.
.' | '.
| | |
'. | .'
'-.__|__. -'
|
x
/
y
Cross-Section for Function of Three Variables
z = k
|
____|____
/ | \
| | |
\____|____/
|
x
/
y
