GRAPHS, LEVEL CURVES ,AND CONTOURS OF FUNCTION OF TWO VARIABLE . LEVEL CURVES OF FUNCTION OF THREE VARIABLES. INTERIOR POINTS, BOUNDARY POINTS.

1. In the preparation of these slides the material has been
taken from online resources inthe shape of books
pictures,research etc. However the instructor dose not
have any intention to take any benefit of these in her
own name.
This lecture is peepared nd dilvered only for
educational purposes.

2. CALCULUS
SABA ALI
BS.Physics
Semester 02

3. • GRAPHS, LEVEL CURVES ,AND CONTOURS OF
FUNCTION OF TWO VARIABLE .
LEVEL CURVES OF FUNCTION OF THREE
VARIABLES.
INTERIOR POINTS, BOUNDARY POINTS.
Goal of this lectures;
 understanding the definitions
 some examples

4. GRAPHS :
A diagram showing the relation between variable quantities,
typically of two variables, each measured along one of a pair
of axes at right angles.
LEVEL CURVE:
The set of points in the plan where a function has a
constant value equal to Z.
𝑧 = 𝑓(𝑥, 𝑦)
with 𝑧 = constant forms a level curve on the surface of 𝑓.

5. CONTOUR :
A level curve projected onto the x y-plane is called a
contour.

8. FUNCTION OF THREE VARIABLES:
In the space the points where a function of three
independent variables has a constant value
f (x , y , z ) = c , make a surface in the
function domain.
LEVEL SURFACES :
The set of points (x , y ,z )in space where a function of
three independent variables has a constant value
f (x , y ,z ) is called level surfaces.

9. Let c =1 we got intercepts (5,0,0) ,(0,4,0)
(0,0,3).

10. 3D graph

11. INTERIOR POINTS AND BOUNDARY POINTS