\begin{equation}
h \mathbf{u} = \left( h u_1 +, \ldots, + h u_n \right)
\end{equation}
References:
[1] W. Kaplan. Advanced Calculus (5th ed.). Addison-Wesley. 2002
Showing posts with label Vector Algebra. Show all posts
Showing posts with label Vector Algebra. Show all posts
Monday, October 3, 2011
Math - Vectors - Scalar multiplication onto a vector
The scalar multiplication onto a vector is simply [1]
\begin{equation}
h \mathbf{u} = \left( h u_1 +, \ldots, + h u_n \right)
\end{equation}
References:
[1] W. Kaplan. Advanced Calculus (5th ed.). Addison-Wesley. 2002
\begin{equation}
h \mathbf{u} = \left( h u_1 +, \ldots, + h u_n \right)
\end{equation}
References:
[1] W. Kaplan. Advanced Calculus (5th ed.). Addison-Wesley. 2002
Math - Vectors - Addition
The sum of two vectors is simply [1]
\begin{equation}
\mathbf{u} + \mathbf{v} = \left( u_1 + v_1 +, \ldots, + u_n + v_n \right)
\end{equation}
References:
[1] W. Kaplan. Advanced Calculus, 5th ed. Addison-Wesley. 2002
[2] A. I. Borisenko and I. E. Tarapov. Vector and Tensor Analysis with Applications, (translated by R. A. Silverman). Dover Publications Inc., Mineola, NY. 1979 (originally published in 1968 by Prentice-Hall, Inc.
\begin{equation}
\mathbf{u} + \mathbf{v} = \left( u_1 + v_1 +, \ldots, + u_n + v_n \right)
\end{equation}
One property of vector addition is known as the commutative property [2]:
\[ \mathbf{A} + \mathbf{B} = \mathbf{B} + \mathbf{A} \]
Another property of vector addition is known as the associative property [2]:
\[ \left( \mathbf{A} + \mathbf{B} \right) + \mathbf{C} = \mathbf{B} + \mathbf{A} \]
References:
[1] W. Kaplan. Advanced Calculus, 5th ed. Addison-Wesley. 2002
[2] A. I. Borisenko and I. E. Tarapov. Vector and Tensor Analysis with Applications, (translated by R. A. Silverman). Dover Publications Inc., Mineola, NY. 1979 (originally published in 1968 by Prentice-Hall, Inc.
Friday, September 23, 2011
Math - Vectors - Dot/scalar/inner product
The dot product, scalar, or inner product is the vector multiplication which executes for two vectors \( \mathbf{A} \) and \( \mathbf{B} \) in three dimensions is [1, 2]
\[ \begin{align} \mathbf{A} \cdot \mathbf{B} &= \left(A_1 \mathbf{e}_1 + A_2 \mathbf{e}_2 + A_3 \mathbf{e}_3 \right) \cdot \left(B_1 \mathbf{e}_1 + B_2 \mathbf{e}_2 + B_3 \mathbf{e}_3 \right) \\ &= A_1 B_1 + A_2 B_2 + A_3 B_3 \end{align} \]
That is, the dot product multiplies each corresponding component of the vectors and adds them together to obtain a scalar. So you could also call this a vector component product.
In progress...to be continued.
References:
\[ \begin{align} \mathbf{A} \cdot \mathbf{B} &= \left(A_1 \mathbf{e}_1 + A_2 \mathbf{e}_2 + A_3 \mathbf{e}_3 \right) \cdot \left(B_1 \mathbf{e}_1 + B_2 \mathbf{e}_2 + B_3 \mathbf{e}_3 \right) \\ &= A_1 B_1 + A_2 B_2 + A_3 B_3 \end{align} \]
That is, the dot product multiplies each corresponding component of the vectors and adds them together to obtain a scalar. So you could also call this a vector component product.
In progress...to be continued.
References:
[1] K. Karamcheti. Principles of Ideal-Fluid Aerodynamics. John Wiley & Sons, Inc., New York, NY. 1966
[2] W. Kaplan. Advanced Calculus (5th ed.). Addison-Wesley. 2002
[2] W. Kaplan. Advanced Calculus (5th ed.). Addison-Wesley. 2002
Math - Vectors - Unit vector and vector magnitude/length/norm
The unit vector of a vector is in the direction of the vector with a magnitude of one. For example, in the three-dimensional Cartesian coordinate system the vectors \( \left( \hat{i}, \hat{j}, \hat{k} \right) \) or \( \left( \mathbf{i}, \mathbf{j}, \mathbf{k} \right) \) represent the unit vectors along the three axes \( \left(x, y, z \right) \).
In general, the unit vector of a vector can be written as
\[ \mathbf{e}_A = \dfrac{ \mathbf{A}}{| \mathbf{A}|} \]
which is the vector of \( A \) divided by the magnitude of \( A \). The magnitude (also called the length or norm) in the three-dimensional case \( \vec{A} = A_1 \mathbf{i} + A_2 \mathbf{j} + A_3 \mathbf{k} \) is expressed as
\[ | \mathbf{A}| = \sqrt{\mathbf{A} \cdot \mathbf{A}} = \sqrt{A_1^2 + A_2^2 + A_3^2} \]
In an orthogonal right-handed system, the unit vectors in three dimensions can be expressed as \( \mathbf{e}_1 \), \( \mathbf{e}_2 \), and \( \mathbf{e}_3 \) where the components of a vector \( \mathbf{A} \) are \( A_1 \), \( A_2 \), and \( A_3 \) and correspond to the unit vectors respectively. That is the vector \( \mathbf{A} \) can be broken down into individual components which relate to whatever coordinate system and unit vectors are chosen to be used [1, 2].
In progress...to be continued.
References:
[1] K. Karamcheti. Principles of Ideal-Fluid Aerodynamics. John Wiley & Sons, Inc., New York, NY. 1966
[2] W. Kaplan. Advanced Calculus (5th ed.). Addison-Wesley. 2002
In an orthogonal right-handed system, the unit vectors in three dimensions can be expressed as \( \mathbf{e}_1 \), \( \mathbf{e}_2 \), and \( \mathbf{e}_3 \) where the components of a vector \( \mathbf{A} \) are \( A_1 \), \( A_2 \), and \( A_3 \) and correspond to the unit vectors respectively. That is the vector \( \mathbf{A} \) can be broken down into individual components which relate to whatever coordinate system and unit vectors are chosen to be used [1, 2].
In progress...to be continued.
References:
[1] K. Karamcheti. Principles of Ideal-Fluid Aerodynamics. John Wiley & Sons, Inc., New York, NY. 1966
[2] W. Kaplan. Advanced Calculus (5th ed.). Addison-Wesley. 2002
Tuesday, September 20, 2011
Vector Analysis and Calculus - General math
This post highlights vector calculus and vector analysis. Such topics and discussions will cover ideas and concepts encountered when dealing with vector analysis. Vectors are a very important concept in science and math and appear very early in calculus and physics classes in college or in advanced high school classes. Vectors are used to describe a direction and magnitude of important concepts in our world such as gravity and other forces. Vectors can also group together many sets of equations into a compact form such as the three-dimensional equations of motion expressed as a single line of a vector equation instead of expanding for each of the three components. Vectors appear very often in fluid dynamics and are a fundamental basis for the subject. Many fluid dynamic books introduce the reader into and with vector analysis [1-5] or contain an appendix at the end of the book [5] while the manuscript by Aris [6] is based upon vectors and fluid dynamics altogether.
As mentioned previously, vectors make up the fundamental basis of mathematics and physics in the world. Many mathematical books which generalize math topics which scientist, engineers, physicists, etc, should be familiar with often place vectorial analysis in the beginning or at least include it in the book type encyclopedia (a collection of topics and methods) of math for applied uses [7].
Vectors not only make up fundamental basis for fluid dynamics, but also for math and physics. In fact it is an underlying foundation in math and physics sciences. Einstein used vector notation to develop and describe his relativity theories. Many in the early stages of the study of math and physics fields generally see vectors in essential courses such as calculus and general physics. In calculus, vector analysis might appear in a II or III semester of the course as the concepts of divergence, curl, gradient, etc. are brought up and discussed. Vectors appear very early at the very beginning in general college physics courses because almost everything is based upon the vector theory. Forces, acceleration, velocity, position, momentum, electrostatics, fluid mechanics and dynamics, etc. all utilize vector analysis.
Specifically, since my background is in fluid dynamics from a graduate aerospace engineering education, I will use my experiences to discuss vectors and other topics. Vectors in fluid dynamics/mechanics can be very useful to describe many fluid concepts and phenomena. Additionally, vectors are very useful as they are coordinate independent or in other words the user does not have to worry about a coordinates system.
A general representation of a vector can be
\[ \vec{A} = A_1 \vec{e}_1 + A_2\vec{e}_2 + A_3 \vec{e}_3 + \ldots + A_n \vec{e}_n = \sum_{i = 1}^n \]
where the coordinate system can be anything and the number of dimensions \( n \). This simple relation is why vectors are so useful, powerful, and why they make up the foundations of our math, physics, and related sciences.
The vector may also be written without the arrow above where boldface font is available.
\[ \mathbf{A} = A_1 \mathbf{e}_1 + A_2 \mathbf{e}_2 + A_3 \mathbf{e}_3 + \ldots + A_n \mathbf{e}_n \]
Scalars according to Serre [ ],
"...are elements of some field \( k \) (or \( K \)), or sometimes of a ring \( R \)."
In progress...
References:
[ ] D. Serre. Matrices: Theory and Applications. Springer-Verlag New York, Inc., New York, NY. 2002
[ ] D. Serre. Matrices: Theory and Applications. 2nd ed. Springer Science+Business Media, LLC., New York, NY. 2010
As mentioned previously, vectors make up the fundamental basis of mathematics and physics in the world. Many mathematical books which generalize math topics which scientist, engineers, physicists, etc, should be familiar with often place vectorial analysis in the beginning or at least include it in the book type encyclopedia (a collection of topics and methods) of math for applied uses [7].
Vectors not only make up fundamental basis for fluid dynamics, but also for math and physics. In fact it is an underlying foundation in math and physics sciences. Einstein used vector notation to develop and describe his relativity theories. Many in the early stages of the study of math and physics fields generally see vectors in essential courses such as calculus and general physics. In calculus, vector analysis might appear in a II or III semester of the course as the concepts of divergence, curl, gradient, etc. are brought up and discussed. Vectors appear very early at the very beginning in general college physics courses because almost everything is based upon the vector theory. Forces, acceleration, velocity, position, momentum, electrostatics, fluid mechanics and dynamics, etc. all utilize vector analysis.
Specifically, since my background is in fluid dynamics from a graduate aerospace engineering education, I will use my experiences to discuss vectors and other topics. Vectors in fluid dynamics/mechanics can be very useful to describe many fluid concepts and phenomena. Additionally, vectors are very useful as they are coordinate independent or in other words the user does not have to worry about a coordinates system.
A general representation of a vector can be
\[ \vec{A} = A_1 \vec{e}_1 + A_2\vec{e}_2 + A_3 \vec{e}_3 + \ldots + A_n \vec{e}_n = \sum_{i = 1}^n \]
where the coordinate system can be anything and the number of dimensions \( n \). This simple relation is why vectors are so useful, powerful, and why they make up the foundations of our math, physics, and related sciences.
The vector may also be written without the arrow above where boldface font is available.
\[ \mathbf{A} = A_1 \mathbf{e}_1 + A_2 \mathbf{e}_2 + A_3 \mathbf{e}_3 + \ldots + A_n \mathbf{e}_n \]
Scalars according to Serre [ ],
"...are elements of some field \( k \) (or \( K \)), or sometimes of a ring \( R \)."
In progress...
References:
[1] K. Karamcheti. Principles of Ideal-Fluid Aerodynamics. John Wiley & Sons, Inc., New York, NY. 1966
[2] R. L. Panton. Incompressible Flow. (3rd ed.). John Wiley & Sons, Inc. Hoboken, NJ. 2005
[3] M. E. O’Neill & F. Chorlton. Viscous and Compressible Fluid Dynamics. Ellis Horwood Limited. Chichester, UK. 1989
[4] T. C. Papanastasiou, G. C. Georgiou, & A. N. Alexandrou. Viscous Fluid Flow. CRC Press. Boca Raton, FL. 2000
[3] M. E. O’Neill & F. Chorlton. Viscous and Compressible Fluid Dynamics. Ellis Horwood Limited. Chichester, UK. 1989
[4] T. C. Papanastasiou, G. C. Georgiou, & A. N. Alexandrou. Viscous Fluid Flow. CRC Press. Boca Raton, FL. 2000
[5] M. T. Schobeiri. Fluid Mechanics for Engineers - A Graduate Textbook. Springer.
Berlin, Germany. 2010
[6] R. Aris. Vectors, Tensors and the Basic Equations of Fluid Mechanics. Dover Publications.
New York, NY. 1990
[7] G. B. Arfken & H.-J. Weber, Mathematical Methods for Physicists (6th ed.). Elsevier Academic Press. Burlington, MA. 2005
[ ] D. Serre. Matrices: Theory and Applications. Springer-Verlag New York, Inc., New York, NY. 2002
[ ] D. Serre. Matrices: Theory and Applications. 2nd ed. Springer Science+Business Media, LLC., New York, NY. 2010
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