Calculus II - 17
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Polar coordinates represent points in a plane using an angle and distance from an origin point, rather than Cartesian x and y distances. In polar coordinates, a point is defined by its radial distance r from the origin and its angle θ measured counterclockwise from the polar axis. Converting between polar and Cartesian coordinates involves simple formulas relating r, θ, x, and y. Polar equations can more simply describe curves like the cardioid than their Cartesian equivalents.
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The polar coordinate system uses a point called the pole and a fixed ray called the polar axis to identify the location of a point P using polar coordinates (r, θ). R represents the distance from the pole to point P, while θ is the angle between the polar axis and a line extending from the pole to point P. Equations in polar coordinates take the form of r = k sinθ or r = j cosθ, where r is the distance and θ is the angle.
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This document discusses cylindrical and spherical coordinate systems. It provides objectives for understanding these coordinate systems, converting between them and Cartesian coordinates, and developing problem-solving skills. Examples are given of converting between cylindrical and Cartesian coordinates, as well as spherical and Cartesian coordinates. Key aspects of cylindrical coordinates include representing points as (r,θ,z) and using conversion equations. Spherical coordinates represent points as (ρ,φ,θ) similar to latitude and longitude. Conversion equations are also provided between the cylindrical and spherical systems.
EMT_2A_cylindrical coordinates.pptx
The document summarizes key concepts from Chapter 1 of the textbook "Engineering Electromagnetics - 8th Edition" by William H. Hayt, Jr. & John A. Buck. It introduces scalar and vector quantities, describes vector algebra including addition, subtraction and multiplication. It also discusses various coordinate systems used to describe the location and direction of vectors including rectangular, cylindrical and spherical coordinate systems. Transformations between Cartesian and other coordinate systems are shown.
Calculus II - 35
The document discusses representations of lines, planes, and space curves in vector calculus. It defines the vector equation of a line, parametric equation of a line, and symmetric equation of a line. It also defines the vector equation and scalar equation of a plane. It then discusses representing space curves using parametric vector functions and calculating derivatives and integrals of vector functions, including rules for differentiation. It provides an example problem proving that the tangent line to a circle at a point is perpendicular to the line connecting that point to the circle's center.
Chapter 12 vectors and the geometry of space merged
This document discusses vectors and geometry in 3D space. It covers topics like 3D coordinate systems, vectors, dot and cross products, equations of lines and planes, cylinders and quadric surfaces. There are also tables listing examples of quadric surface graphs. The document provides information on representing and analyzing geometric objects in 3D space using vectors and coordinate systems.
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This document discusses different coordinate systems used to describe spatial variations of physical quantities in electromagnetism, including Cartesian, cylindrical, and spherical coordinates. It provides the definitions and relationships between the variables in each system. In Cartesian coordinates, a point P is defined by coordinates (x,y,z). In cylindrical coordinates, a point P is defined by (p,φ,z) where p is the radial distance, φ is the azimuthal angle, and z is the same as in Cartesian. In spherical coordinates, a point P is defined by (r,θ,φ) where r is the radial distance, θ is the colatitude angle, and φ is the azimuthal angle. Transformations between the different coordinate
Calculus II - 36
The document discusses arc length, curvature, and formulas for calculating them for curves in 2D and 3D. It provides the definitions of arc length, unit tangent vector, and curvature. Examples are given of finding the curvature of various curves, such as circles, helices, and parabolas, at given points using parametrizations of the curves. In general, the curvature of a plane curve f(x) = y is k(x) = f''(x) / [1+(f'(x))^2]^(3/2).
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This document discusses polar coordinate systems and their relationship to Cartesian coordinates. It provides examples of plotting points, graphing polar equations, and converting between polar, Cartesian, and parametric representations of curves. Key topics covered include the polar coordinate system, converting between polar and Cartesian coordinates, graphing common polar curves like circles, cardiods, and lemniscates, classifying polar equations, and representing curves parametrically.
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A parabola is a conic section defined as the set of all points in a plane that are equidistant from a fixed point called the focus and a line called the directrix. The general equation of a parabola is y=ax^2, where a is a constant related to the distance between the focus and directrix. Given a focus point and directrix, every point on the parabola satisfies the property that its distance to the focus is equal to its distance to the directrix. Parabolas can be used to model projectile motion under gravity.
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The document discusses key concepts related to the Cartesian plane including:
- The Cartesian plane is formed by two perpendicular number lines called the x-axis and y-axis intersecting at the origin point.
- Any point P on the plane can be located using its coordinates (x,y) which indicate the point's position along the x and y axes.
- The distance between two points P1(x1,y1) and P2(x2,y2) can be calculated using the distance formula.
- Key curves that can be represented on the Cartesian plane include lines, circles, parabolas, ellipses, and hyperbolas through their defining equations.
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In this Presentation, I have explained the co-ordinate system in three plain. ie Cylindrical, Spherical, Cartesian(Rectangular) along with its Differential formulas for length, area &volume.
Vectors.pdf
The document defines a vector as having both magnitude and direction, represented geometrically by an arrow. It discusses representing vectors algebraically using coordinates, and defines operations like addition, subtraction, and scaling of vectors. Key vector concepts covered include the dot product, which yields a scalar when combining two vectors, and unit vectors, which have a magnitude of 1. Examples are provided of using vectors to solve problems and prove geometric properties.
Vectors.pdf
The document defines vectors and discusses their geometric and algebraic representations. Geometrically, a vector has a magnitude and direction represented by an arrow. Algebraically, a vector in a plane can be represented by its coordinates (a1, a2) and in 3D space by coordinates (a1, a2, a3). Vectors can be added by placing them head to tail, subtracted by reversing one and adding, and scaled by a scalar number. The dot product of two vectors A and B yields a scalar value that geometrically equals the magnitudes of A and B multiplied by the cosine of the angle between them.
Vectors.pdf
The document defines vectors and discusses their geometric and algebraic representations. Geometrically, a vector has a magnitude and direction represented by an arrow. Algebraically, a vector in a plane can be represented by its coordinates (a1, a2) and in 3D space by coordinates (a1, a2, a3). Vectors can be added by placing them head to tail, subtracted by reversing one and adding, and scaled by a scalar number. The dot product of two vectors A and B yields a scalar equal to |A||B|cosθ, where θ is the angle between the vectors.
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Chapter 12 vectors and the geometry of space merged
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The cross product and dot product of vectors are introduced. Properties of the cross product include that it is only defined for 3D vectors and that the magnitude of the cross product equals the area of the parallelogram determined by the two vectors. Vector, parametric, and various forms of plane equations are presented along with examples of finding equations of lines and planes given conditions.
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The dot product of two vectors a and b is defined as a·b and is calculated by multiplying their corresponding components and adding them. It can be used to calculate the projection of one vector onto another and the cosine of the angle between the vectors. The cross product of two vectors a and b is defined as axb and produces a third vector that is perpendicular to both a and b. It has various properties including being anticommutative and having a magnitude equal to the area of the parallelogram determined by the two vectors.
Calculus II - 32
A vector is a quantity that has both magnitude and direction. It can be represented in a coordinate system with points (x,y,z) and the vector between two points is defined as (x2-x1, y2-y1, z2-z1). Vectors can be added, subtracted, and multiplied by scalars. The length of a vector is calculated using the Pythagorean theorem and vectors can be unitized. There are standard basis vectors i, j, k in 3D space and vectors can be represented as a combination of these basis vectors. The dot and cross products can be used to calculate properties of vectors.
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The document discusses 3D coordinate systems and vectors. It defines:
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The document discusses using Taylor series approximations to evaluate expressions. It finds the maximum error of approximating e-0.5 as 1 - 0.5 + 0.5^2, determines the value of x for which this approximation is accurate to within 0.01, and establishes that an 8th degree Taylor polynomial is needed to approximate e^x in [-1,1] with an error less than 0.001.
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The document discusses Taylor series and Maclaurin series. It provides the formulas for determining the coefficients of a Taylor series centered at a point a. It also discusses Taylor's inequality which relates the remainder of a Taylor series to the derivative of the function. Several examples are worked out of finding the Maclaurin series for common functions and proving they represent the functions for all values. Additional power series are derived from fundamental series like e^x, sin(x), and cos(x).
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The document discusses Taylor series and how they can be used to represent general functions as power series. It provides the formulas for determining the coefficients in the Taylor series expansion of a function f(x) about a point a. Examples are given of finding the Maclaurin series, which is the Taylor series expansion about x=0, for common functions and determining their radii of convergence. The key steps in proving that the Taylor series equals the original function within the radius of convergence are outlined using Taylor's inequality.
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This document discusses power series representations of functions and their radii of convergence. It provides examples of finding the power series for functions like 1/(1+x) and sin(x), as well as the general process for determining the power series representation of a function and the radius of convergence based on the interval of convergence. It also discusses rules for differentiating and integrating term-by-term for power series whose functions have a given radius of convergence.
Calculus II - 26
A power series is a series of the form Σan(x-c)n where an are the coefficients of the series. The sum of the series is a function of x. A power series centered at c or about c is of the form Σan(x-c)n. For a given power series, there are three possibilities for convergence: it converges only at c, converges for all x, or converges when |x-c|<R and diverges when |x-c|>R, where R is the radius of convergence.
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This document discusses tests for determining if infinite series converge or diverge, including:
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Calculus II - 24
The Comparison Test allows determining if a series converges or diverges by comparing it to standard p-series or geometric series. If a series is less than or equal to a convergent p-series, or greater than or equal to a divergent p-series, then the series has the same behavior. Examples demonstrate applying the test to various series.
Calculus II - 23
The document introduces the integral test, which can be used to determine if an infinite series is convergent or divergent. The integral test states that a series is convergent if the corresponding improper integral converges, and divergent if the improper integral diverges. Several examples demonstrate applying the integral test to various series by setting up and evaluating the corresponding improper integrals. The document also discusses using integrals to estimate the sum of a convergent series.
Calculus II - 22
This document discusses convergence of infinite series. It defines partial sums of a series and states that if the sequence of partial sums converges, then the series converges to that limit. If the partial sums do not converge or their limit does not exist, then the series diverges. It provides examples of geometric series, which converge if the common ratio is less than 1 in absolute value, and harmonic series, which diverges. It also discusses using known convergent series to determine if new series formed from their sums or products converge.
Calculus II - 21
This document discusses infinite sequences and infinite series. Some key points:
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Calculus II - 19
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Calculus II - 16
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Calculus II - 17
- 1. 10.3 Polar Coordinates Cartesian coordinates: a point in the plane is represented by the distances from two perpendicular axes. y o x
- 2. Polar coordinates system: O - pole (or origin) Polar axis: a (horizontal) ray starting at O Any point P is represented by ( , ) . P r O Polar axis
- 3. Conventions: Pole is represented by ( , ) for any . is measured in the counter-clockwise direction. ( , ) represents the same point as ( , ). (− , ) represents the same point as (, + ). In polar coordinate system each point has many representations.
- 4. Connection between polar and Cartesian coordinates system: = , = = + , = Converting polar coordinates to Cartesian coordinates is easy, while representing the point with Cartesian coordinates in terms of polar coordinates is harder.
- 5. = , = = + , = Ex: Convert ( , / ) from polar to Cartesian coordinates, convert ( , ) from Cartesian to polar coordinates.
- 6. = , = = + , = Ex: Convert ( , / ) from polar to Cartesian coordinates, convert ( , ) from Cartesian to polar coordinates. ( , / ) ( , )
- 7. = , = = + , = Ex: Convert ( , / ) from polar to Cartesian coordinates, convert ( , ) from Cartesian to polar coordinates. ( , / ) ( , ) ( , / ) ( , )
- 8. = , = = + , = Ex: Convert ( , / ) from polar to Cartesian coordinates, convert ( , ) from Cartesian to polar coordinates. ( , / ) ( , ) ( , / ) ( , ) not ( , / )!
- 9. Polar curves. Like in the Cartesian system, one polar equation represents a curve in general. Sometimes it is much simpler than the corresponding representation in the Cartesian system. Ex: What is the curve described by = ? What is the curve described by = ?
- 10. Ex: =
- 11. Ex: =
- 12. Ex: =
- 13. Ex: = + (cardioid)
- 14. Ex: = + (cardioid)
- 15. Ex: = + (cardioid)
- 16. Ex: =
- 17. Ex: =
- 18. Ex: =
- 19. Ex: =
- 20. Ex: = ( . )+ ( . )
- 21. Ex: =
- 22. Ex: = ( . )+ ( )
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