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Old 08-03-2007   #1
Jim M
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Default Advance Math: Linear Algebra + Vector ****ysis?

I took Calculus in High School and College. I liked Math a lot and wanted to go farther. I took Linear Algebra and then Vector ****ysis. However, I don't feel I mastered that material. For example, I didn't understand Green's Theorem, or Stoke's Theorem. I also didn't fully understand what an eigen-value was. I passed the classes, because I did the homework, and memorized the problem solving, but I didn't understand the theory behind them. I think it was because my professors for those classes, didn't fully understand English. I would really like to self-teach myself those concepts. What is the best way to do so?
 
Old 08-03-2007   #2
Nostradamus is back
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get stoned and study it
 
Old 08-03-2007   #3
Payam
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I am a math instructorand I intend to help you learning these matters.Believe me, it is very simple.First of all you must be sure that you know what are these objects1) a scalar field (also called a multivariable scalar function)It is a function f:R2->R or f:R3->Rthus f(x,y) is a 2 dimensional scalar fieldand f(x,y,z) is a 3 dimensional scalar field.(2) a vector fieldIt is a funtion f:R2->R2 or f:R3->R3It assigns a vector to each point of the space.like (1) it may be 2-dimensional or 3-dimensional(3) The cross product of vectorsAlthough it appears as the most innocent item of the listIT IS THE MAIN DIFFICULTY!!!Perhaps you know exactly what is the cross product of two 3-dimensional vectors? yes, it is a 3-dimensional vector.But I'm sure you cannot imagine what the cross product of two 2-dimensional vectors is. Unbelievably enough it is a SCALAR!!!let u=u1*i+u2*j and v=v1*i+v2*jthen u cross v is defined as u1*v2-u2*v1.(4) the curl of a vector fieldIt is defined as (nabla cross f)It is a vector field when f is 3-dimensionalIT IS A SCALAR FIELD WHEN f IS 2-DIMENSIONAL!!!(4) the double integral of a 2-dimensional scalar field over a 2 dimensional domain in R2(5) the work integral or the second kind line integral of a vector field over a curveIt is also of two types:2-dimensional and 3-dimensionallet f be a 2-dimensional vector field defined over Dwhere D is a bounded 2-dimensional subset of R2 with a boundarycurl(f) is a 2-dimensional scalar field defined on DGREEN's THEOREM:double integral of curl(f) over Dis equal to theline integral of f over the boundary of CAs you see there are many kinds of integrals involved.and they have many differences with the definite integral which you know.THE STOKES THEOREMis about summing up the curl(f) in 3-dimensions over a surface, using what is called a flux integral or a surface integral of the second kind. If the surface has a boundary C then the theorem states that it is equal to the work integral of f over the C. (check the types of fields involved)As you can see the stokes' theorem is the 3 dimensional generalization of Green's theorem.GAUSS THEOREMis about summing up the div(f) over a 3 dimensional volume V, it is equal to the flux integral of f over the boundary of V.(again check the type of fields)Believe me!! If you know what the integrals are then you can simply prove them all by almost direct computation.Eigenvalues are not important for the 2nd year calculus.Just an example can show you its importance. Each rotation in space have an axis. All the vectors in space are moved by this transform except those which rely on the axis of rotation. They are eigenvectors of that linear transformation.NOW it is the time to stick to your textbooks.FINALLY excuse my English ;o)
 
Old 08-03-2007   #4
LALIT B
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Try This : "Div, Grad, Curl" by H. M. Schey or "Geometric Vectors" by Gabriel Weinreich . . . those are the best texts on vector ****ysis . . . "Linear Algebra Done Right" by Sheldon Axler and "Linear Algebra Demystified" by David McMahon should teach you all you need to know about Linear Algebra.(1) a scalar field (also called a multivariable scalar function)It is a function f:R2->R or f:R3->Rthus f(x,y) is a 2 dimensional scalar fieldand f(x,y,z) is a 3 dimensional scalar field.(2) a vector fieldIt is a funtion f:R2->R2 or f:R3->R3It assigns a vector to each point of the space.like (1) it may be 2-dimensional or 3-dimensional(3) The cross product of vectorsAlthough it appears as the most innocent item of the listIT IS THE MAIN DIFFICULTY!!!Perhaps you know exactly what is the cross product of two 3-dimensional vectors? yes, it is a 3-dimensional vector.But I'm sure you cannot imagine what the cross product of two 2-dimensional vectors is. Unbelievably enough it is a SCALAR!!!let u=u1*i+u2*j and v=v1*i+v2*jthen u cross v is defined as u1*v2-u2*v1.(4) the curl of a vector fieldIt is defined as (nabla cross f)It is a vector field when f is 3-dimensionalIT IS A SCALAR FIELD WHEN f IS 2-DIMENSIONAL!!!(4) the double integral of a 2-dimensional scalar field over a 2 dimensional domain in R2(5) the work integral or the second kind line integral of a vector field over a curveIt is also of two types:2-dimensional and 3-dimensionallet f be a 2-dimensional vector field defined over Dwhere D is a bounded 2-dimensional subset of R2 with a boundarycurl(f) is a 2-dimensional scalar field defined on DGREEN's THEOREM:double integral of curl(f) over Dis equal to theline integral of f over the boundary of CAs you see there are many kinds of integrals involved.and they have many differences with the definite integral which you know.THE STOKES THEOREMis about summing up the curl(f) in 3-dimensions over a surface, using what is called a flux integral or a surface integral of the second kind. If the surface has a boundary C then the theorem states that it is equal to the work integral of f over the C. (check the types of fields involved)As you can see the stokes' theorem is the 3 dimensional generalization of Green's theorem.GAUSS THEOREMis about summing up the div(f) over a 3 dimensional volume V, it is equal to the flux integral of f over the boundary of V.(again check the type of fields)Believe me!! If you know what the integrals are then you can simply prove them all by almost direct computation.Eigenvalues are not important for the 2nd year calculus.Just an example can show you its importance. Each rotation in space have an axis. All the vectors in space are moved by this transform except those which rely on the axis of rotation. They are eigenvectors of that linear transformation.
 
 

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