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Why Vector Calculus would be a perquisite or a co-requisite?

  • Thread starter Nusc
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Could anyone tell me why Vector Calculus would be a prequisite or a co-requisite? Specifically, what topics are required to know vector calculus in PDE's?

I suspected a course in ODE's would be enough.

Thanks
 

HallsofIvy

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PDEs necessarily involve multi-variable calculus- that is most naturally set in terms of vector calculus.
 

robphy

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Note that the laplacian-operator which appears in, e.g., the wave, poisson, and heat PDEs is div(grad( )), which involve operations developed in a vector calculus course.
 

dextercioby

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Actually everything comes from differential geometry.U'll see that the laplacian is the trace of the hessian and it can be written in terms of differentials and codifferentials.

Vector calculus is not compulsory for PDE-s.Linear algebra,multivariable calculus (partial derivatives),complex analysis (and special functions) and ODE-s are.

Daniel.
 

robphy

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dextercioby said:
Actually everything comes from differential geometry.U'll see that the laplacian is the trace of the hessian and it can be written in terms of differentials and codifferentials.

Vector calculus is not compulsory for PDE-s.Linear algebra,multivariable calculus (partial derivatives),complex analysis (and special functions) and ODE-s are.

Daniel.
It's fair to say that vector-calculus is a pre-requisite for differential geometry.

While it may not be compulsory, there's much to be gained to have a vector-calculus background for a large class of physically-interesting PDEs and systems-of-PDEs. For instance, the Maxwell Equations (written by Maxwell as a system of PDEs [20 eqs and 20 unknowns]) benefited from its reformulation in terms of vector-calculus by Heaviside et al. For one thing, one has a geometric interpretation with which one can seek and exploit symmetries. Imagine the Einstein field equations (a larger system of PDEs) without the aid of vector/tensor-calculus.
 
i heard that if you're not a math major, then the toughest courses are:
1)dynamics&chaos
2)pde
 
M

MalleusScientiarum

dynamics and chaos is probably easier if you have a solid physics background, since I've found that a lot of mathematicians don't like nonlinear things since they're really hard to define and study in general.
 
i dont know if that course requires any physics knowledge, and i dont know any physics. but ive been told that people that are naturally good at physics are good at math and viceversa. Hence, a course called differential geometry, which physicists take, i do not consider it a non-math course. on the other hand, pde and dynamics&chaos is also taken by some engineers and in my major , which is like theoretical biology. At my school, for pde the prereqs are ode,vector calc, and linear algebra. and for dynamics&chaos only linear algebra. but there is dynamics&chaos2 and that one requires analysis3 as a prereq
 
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lurflurf

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Nusc said:
Could anyone tell me why Vector Calculus would be a prequisite or a co-requisite? Specifically, what topics are required to know vector calculus in PDE's?

I suspected a course in ODE's would be enough.

Thanks
There are several reasons
1) Dumb prerequisite like the algebra classes with vector calc prerequisites.
2) At that school some of the needed knowlege is covered in vector calc, at many schools vector calc includes essential matiarial on the calculus of several variables.
3)As some mentioned many partial differential equations are written using vector calculus symbols, but this is a poor reason as notation can be explained, and is not essential. In most first courses one need not concentrate much on equations involving lots of complicated equations anyway. There is plenty to do with simple equations like 1st order equations and 1d heat,wave,laplace,hemholtz.
 
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In the first semester of 2nd college year comes linear algebra, ODEs and calculus III (or multivariable calculus). Basic differential geometry is only for the 3rd year.
 

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