(MTH 250) Lecture 26 Calculus. Previous Lecture’s Summary Recalls Introduction to double integrals Iterated integrals Theorem of Fubini Properties of.

(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.