(MTH 250)

Lecture 26

Calculus

Previous Lecture’s Summary

•Recalls

•Introduction to double integrals

•Iterated integrals

•Theorem of Fubini

•Properties of double integrals

•Integrals over non-rectangular regions

•Reversing the order of integration

Today’s Lecture

•Recalls

•Polar Coordinates

•Rectangular Coordinates.

•Cylindrical Coordinates

•Spherical Coordinates

•Equations of Surfaces

•Conversion of Coordinate Systems

•Directional Derivatives

•Gradients.

• Using linesparallel to the coordinate axes, divide the rectangle enclusing the regionintosubrectangles.

• Chooseanyarbitrary point in eachsubrectangle.

• Let denote the area of the rectangle.

• The volume of a rectangularparallelopipedwith base area and heightisgiven by

• Approximation to the volume of the entiresloidis.

Recalls

Definition:

Definition: The double integral of a function over a regionisdefined as the limit of the Riemann sums and isdenoted by

Recalls

Definition:

Definition:

Recalls

Theorem:

Recalls

Definition:

Recalls

Properties of double integralsTheorem:

Polar coordinates

Definition: Polar Coordinates are two values that locate a point on a plane by its distance from a fixed pole and its angle from a fixed line passing through the pole.

Let be a point in plane. Then using trigonometry we have

Polar coordinates

Definition: Let be a point in polar coordinate plane. Again by using trigonometry we have

.

.

Polar coordinates

Examples: Consider the point inplane. In Cartesiancoordinatesthisbecomes

The point in plane canbeconverted to plane as

2,322

14,

2

34

6,4 So

2

1

6sin ,

2

3

6cos

64 ,

6cos4

6,4

cin

6.1121804.67

4.675

12tan

5

12tan 1

13169

14425125 222

r

r

6.112,13)12,5(

Rectangular coordinates

• Three coordinates are required to establist the location of a point in .

• This wecan do using the rectangularcoordinates of a point where and are respectively the displacementsalongand axis respectively.

• The coordinatescanbeany real numbers, withoutany restriction.

Cylindrical coordinates

A point in canberepresented by threequantities

• Distance from the origin, • Angle with the polar-axis,• Heightabove the plane.

This coordinate system iscalled the Cylindricalcoordinate system.

There are restrictions on the allowable values of the coordinates.

Cylindrical coordinates

The cylindricalcoordinatesjustadd a coordinate to the the polar coordinates

Consider a point in cylindricalcoordinates. Then, in rectangularcoordinates

,

The point in is given by

Cylindrical coordinates

A point in canberepresented by threequantities

• Distance from the origin , • Angle with the polar-axis,• Angle with the z-axis.

This coordinate system iscalled the Sphericalcoordinate system.

There are restrictions on the allowable values of the coordinates.

Spherical coordinates

Spherical coordinates

Cartesian to Sphericalcoordinates:

Spherical coordinates

Spherical coordinates

Equations of surfaces

Conversion of Coordinate systems

Directional Derivatives

Directional Derivatives

Theorem:

Directional Derivatives

Directional Derivatives

Directional Derivatives

GradientDefinition:

Remark:

GradientProperties:

,)( VUVU

,)( UVVUUV

,)( 1 VnVV nn

Gradient

Gradient

GradientRemarks:

• Let be a point on a levelcurve and the curvecanbesmoothlyparametrized as

• The the tangent vectoris.

• Differentiatiing the levelcurveweobtain

• gives a direction alongwhichisnearly constant and so

GradientTheorem:

Remark:

Gradient

Gradient

Lecture Summary

•Polar Coordinates

•Rectangular Coordinates.

•Cylindrical Coordinates

•Spherical Coordinates

•Equations of Surfaces

•Conversion of Coordinate Systems

•Directional Derivatives

•Gradients.