Lecture-2
Mathematics 1 (15B11MA111)
CO [C105.1]
Module: Partial differentiation
1
,Topics to be covered
Function of two variables
Neighborhood of a point
Limit
Definition
Examples
Assignment
Reference for the lecture
R.K Jain and S.R.K. Iyenger, “Advanced Engineering Mathematics” fifth
edition, Narosa publishing house, 2016.
2
,Function of two variables
Let 𝒛 = 𝒇 𝒙, 𝒚 be a function of two variables then the set of points
(𝒙, 𝒚) for which 𝒛 𝒐𝒓 𝒇(𝒙, 𝒚) be defined is called domain of 𝒛 and
the corresponding values of 𝒛 is called range of 𝒛.
For example: (i) Let 𝒛 = 𝟏 − 𝒙𝟐 − 𝒚𝟐 and 𝒛 𝒊𝒔 𝒓𝒆𝒂𝒍.
Then 𝟏 − 𝒙𝟐 − 𝒚𝟐 ≥ 𝟎, 𝒊. 𝒆 𝒙𝟐 + 𝒚𝟐 ≤ 𝟏, which is its domain and
the range is the set of all real positive numbers.
𝟏
(ii) Let 𝒛 = .
𝒙−𝒚
Here domain is the set of all points for which 𝒙 ≠ 𝒚 and the range
is R.
3
, Distance between two points
The distance between the points 𝑨 𝒙𝟎 , 𝒚𝟎 𝒂𝒏𝒅 𝑩 𝒙𝟏 , 𝒚𝟏 is given
as 𝒅 𝑨, 𝑩 = 𝒙𝟏 − 𝒙𝟎 𝟐 + 𝒚 𝟏 − 𝒚𝟎 𝟐
Neighborhood of a point:
Let 𝑷(𝒙𝟎 , 𝒚𝟎 ) be a point in 2D
Space. The (𝜹 − 𝒏𝒆𝒊𝒈𝒉𝒃𝒐𝒓𝒉𝒐𝒐𝒅) 𝒐𝒇
𝑷(𝒙𝟎 , 𝒚𝟎 ) is the set of all points which lies inside
the circle of radius 𝜹 with center at 𝑷(𝒙𝟎 , 𝒚𝟎 ).
4
Mathematics 1 (15B11MA111)
CO [C105.1]
Module: Partial differentiation
1
,Topics to be covered
Function of two variables
Neighborhood of a point
Limit
Definition
Examples
Assignment
Reference for the lecture
R.K Jain and S.R.K. Iyenger, “Advanced Engineering Mathematics” fifth
edition, Narosa publishing house, 2016.
2
,Function of two variables
Let 𝒛 = 𝒇 𝒙, 𝒚 be a function of two variables then the set of points
(𝒙, 𝒚) for which 𝒛 𝒐𝒓 𝒇(𝒙, 𝒚) be defined is called domain of 𝒛 and
the corresponding values of 𝒛 is called range of 𝒛.
For example: (i) Let 𝒛 = 𝟏 − 𝒙𝟐 − 𝒚𝟐 and 𝒛 𝒊𝒔 𝒓𝒆𝒂𝒍.
Then 𝟏 − 𝒙𝟐 − 𝒚𝟐 ≥ 𝟎, 𝒊. 𝒆 𝒙𝟐 + 𝒚𝟐 ≤ 𝟏, which is its domain and
the range is the set of all real positive numbers.
𝟏
(ii) Let 𝒛 = .
𝒙−𝒚
Here domain is the set of all points for which 𝒙 ≠ 𝒚 and the range
is R.
3
, Distance between two points
The distance between the points 𝑨 𝒙𝟎 , 𝒚𝟎 𝒂𝒏𝒅 𝑩 𝒙𝟏 , 𝒚𝟏 is given
as 𝒅 𝑨, 𝑩 = 𝒙𝟏 − 𝒙𝟎 𝟐 + 𝒚 𝟏 − 𝒚𝟎 𝟐
Neighborhood of a point:
Let 𝑷(𝒙𝟎 , 𝒚𝟎 ) be a point in 2D
Space. The (𝜹 − 𝒏𝒆𝒊𝒈𝒉𝒃𝒐𝒓𝒉𝒐𝒐𝒅) 𝒐𝒇
𝑷(𝒙𝟎 , 𝒚𝟎 ) is the set of all points which lies inside
the circle of radius 𝜹 with center at 𝑷(𝒙𝟎 , 𝒚𝟎 ).
4
