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Three-Dimensional Coordinate Systems - USM

Jim Lambers MAT 169. Fall Semester 2009-10. Lecture 17 Notes These notes correspond to Section in the text. Three-Dimensional Coordinate Systems Over the course of the next several lectures, we will learn how to work with locations and directions in Three-Dimensional space, in order to easily describe objects such as lines, planes and curves. This will set the stage for the study of functions of two variables, the graphs of which are surfaces in space. Points in Three-Dimensional Space Previously, we have identi ed a point in the -plane by an ordered pair that consists of two real numbers, an - Coordinate and - Coordinate , which denote signed distances along the -axis and -axis, respectively, from the origin, which is the point (0, 0). These axes, which are collectively referred to as the Coordinate axes, divided the plane into four quadrants. We now generalize these concepts to Three-Dimensional space, or -space.

Three-Dimensional Coordinate Systems Over the course of the next several lectures, we will learn how to work with locations and directions in three-dimensional space, in order to easily describe objects such as lines, planes and curves. This will set the stage for the study of functions of two variables, the graphs of which are surfaces in space.

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Three-Dimensional Coordinate Systems

www.math.usm.edu

Jim Lambers MAT 169 Fall Semester 2009-10 Lecture 17 Notes These notes correspond to Section 10.1 in the text. Three-Dimensional Coordinate Systems

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Orthogonality of Bessel Functions - USM

www.math.usm.edu

Normalization Now that we have orthogonal Bessel functions, we seek orthonormal Bessel functions. From Z a 0 ˆ[J (kˆ)]2 dˆ= lim k0!k a[k 0J (ka)J (ka) kJ (ka)J (ka)]

  Functions, Orthogonal, Bessel functions, Bessel, Orthogonal bessel functions

Properties of Sturm-Liouville Eigenfunctions and Eigenvalues

www.math.usm.edu

Real Eigenvalues Just as a symmetric matrix has real eigenvalues, so does a (self-adjoint) Sturm-Liouville operator. Proposition 2 The eigenvalues of a regular or periodic Sturm-Liouville problem are real.

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The Secant Method - USM

www.math.usm.edu

Jim Lambers MAT 772 Fall Semester 2010-11 Lecture 4 Notes These notes correspond to Sections 1.5 and 1.6 in the text. The Secant Method One drawback of Newton’s method is that it is necessary to evaluate f0(x) at various points, which may not be practical for some choices of f.

  Lecture, Notes, Methods, Ascent, Secant method, Lecture 4 notes

Gram-Schmidt Orthogonalization - USM

www.math.usm.edu

each polynomial depends on the previous two. Table lists several families of orthogonal polynomials that can be generated from such a recurrence relation; we will see some of these families later in the course. Polynomials Scalar Product Legendre R 1 1 P n(x)P m(x)dx= 2 mn=(2n+ 1) Shifted Legendre R 1 0 P n(x)P m (x)dx= mn=(2n+ 1) Chebyshev ...

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The Wronskian - math.usm.edu

www.math.usm.edu

We de ne a second-order linear di erential operator Lby L[y] = y00+ p(t)y0+ q(t)y: Then, a initial value problem with a second-order homogeneous linear ODE can be stated as L[y] = 0; y(t 0) = y 0; y0(t 0) = z 0: We state a result concerning existence and uniqueness of solutions to such ODE, analogous to the Existence-Uniqueness Theorem for rst ...

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Linear Interpolating Splines - USM

www.math.usm.edu

Linear Interpolating Splines We have seen that high-degree polynomial interpolation can be problematic. However, if the tting function is only required to have a few continuous derivatives, then one can construct a piecewise polynomial to t the data. We now precisely de ne what we mean by a piecewise polynomial.

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Gaussian Elimination and Back Substitution

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The process of eliminating variables from the equations, or, equivalently, zeroing entries of the corresponding matrix, in order to reduce the system to upper …

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Properties of Sturm-Liouville Eigenfunctions and …

www.math.usm.edu

The following essential result characterizes the behavior of the entire set of eigenvalues of Sturm-Liouville problems. Proposition 6 The set of eigenvalues of a regular Sturm-Liouville problem is countably in nite, and is a monotonically increasing sequence 0 < 1 < 2 < < n< n+1 < with lim n!1 n = 1. The same is true for a periodic Sturm ...

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Orthogonality of Bessel Functions - USM

www.math.usm.edu

Orthogonality of Bessel Functions Since Bessel functions often appear in solutions of PDE, it is necessary to be able to compute coe cients of series whose terms include Bessel functions. Therefore, we need to understand their orthogonality properties. Consider the Bessel equation ˆ2 d2J (kˆ) dˆ2 + ˆ dJ (kˆ) dˆ + (k2ˆ2 2)J (kˆ) = 0 ...

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