(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.
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