Math 275 – May 10 (Final Notes)

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All of the points mentioned below are covered in your book and in the online class notes – with examples. Of course everything was covered with examples in class as well. You have plenty of homework assigned so I will not suggest more here. for examples look at the other online notes and in the text.

1. Line integrals

Definition of path, curve, simple path/curve, closed path/curve, Jordan path/curve, smooth curve, and piecewise smooth curve.
You need to know what ${\int_C f\cdot \vec T\,ds}$ and ${\int_C f\cdot\vec n\,ds}$ mean and how to calculate these.
You should know how to use line integrals to calculate mass, average value of ${f}$ along ${C}$ , center of mass, moments of inertia, etc.
You need to understand the notion of orientation of a curve.
You need to understand and know how to calculate, flux through a curve, flow along a curve, and work along a curve.
You need to know what circulation is and what the notation ${\oint_C f\,ds}$ means. You need to know what the standard (outward) orientation of a Jordan curve is.

2. Del

You should know what ${\nabla}$ is and what ${\nabla\cdot F}$ (divergence of ${F}$ ), ${\nabla \times F}$ (curl of ${F}$ ) are.
You should know some trivial facts: ${\nabla\times(\nabla f)=0}$ (since this is just ${(\nabla\times\nabla)f}$ ) so in particular the curl of the gradient is always ${0}$ . ${\nabla\cdot(\nabla\times F)=0}$ – recall ${\vec v\cdot(\vec u\vec w)}$ is the triple product and is the same as ${\det(\vec u,\vec v,\vec w)}$ , so ${\det(\nabla,\nabla,F)=0}$ since the determinant of a matrix with two identical rows is ${0}$ . So the divergence of the curl is always ${0}$ .

3. Conservative fields

Definition of conservative field as well as restrictions on the domain so that the definition makes sense, i.e., open and path connected.
Path independence, equivalence of path independence with conservative.
Gradient fields/potential functions, equivalence with conservative fields under certain assumptions.
Fundamental theorem for line integrals, ${\int_C F\,dr=f(B)-f(A)}$ where ${F}$ is a conservative field defined on an open path connected region ${R}$ , ${C}$ is a piecewise smooth curve from ${A}$ to ${B}$ lying in ${R}$ , and ${F=\nabla f}$ for some potential function ${f}$ .
Test to check when a field is conservative, including conditions on when the test works. (Open, path connected, simply connected region; ${F}$ has continuous second partials.)
Be able to find a potential function for a conservative field and use this to evaluate a line integral.

4. Surface integrals

You need to know what a smooth surface is and how they fit together to form piecewise smooth surfaces.
You need to know and understand what an orientation of a smooth/piecewise smooth surface is and how to find an orientation if it exists.
You need to know what surface integrals are and how to compute them. In particular you need to know what ${\iint_S f\,d\sigma}$ means, how to interpret ${d\sigma}$ – in the notes there are several ways of interpreting ${d\sigma}$ mentioned.
You should know how to apply surface integrals to find surface area, mass of a thin surface, center of mass of a thin surface, moment of inertia of a thin surface, average value of ${f}$ on ${S}$ , etc.
You should know the definition of flux of ${F}$ through ${S}$ and how to compute this.

5. The fundamental theorems of calculus – Green’s, Stoke’s, Divergence

You should know and be able to use the normal and tangential forms of Green’s theorem in the plane. In particular you must know the hypotheses of Green’s theorem and when it applies. You must know what the induced orientation of the boundary of ${R}$ is and how to compute it.
You should be able to use Green’s to simplify the calculation of certain line integrals and conversely to compute area of planar regions.
You should have some idea of how Green’s gives the test for path independence and why simple connectedness is important.
You should understand the way Green’s theorem can be used in the case that ${\nabla\cdot F=0}$ or ${(\nabla\times F)\cdot \hat k=0}$ . For example given a field ${F}$ with ${0}$ divergence on a region ${R}$ and a Jordan curve ${C}$ in ${R}$ , then ${\oint_C F\cdot\vec n\,ds=\oint_{C'} F\cdot\vec n\,ds}$ where ${C'}$ might be taken to be a much simpler curve. (This might be the point of the problem, it is up to you to pick a simple ${C'}$ and evaluate the line integral there.)
Stoke’s theorem is essentially just the tangential form of Green’s theorem for arbitrary surfaces satisfying certain conditions – that you must know.
You should be able to use Stoke’s theorem to calculate the flux of ${\nabla\times F}$ through some surface via computing a simple line integral on the boundary.
You should know why you can replace a given surface by a much simpler one having the same boundary and use this in computations of ${\iint_S\nabla\times F\,\vec n\,d\sigma}$ . For example if I ask you to compute the flux of ${\nabla\times F}$ through the parabolic surface ${S}$ ${z=x^2+y^2}$ from ${z=0}$ to ${z=2}$ . Then you know ${\int_S \nabla\times F\vec n\,ds=\int_{S'}\nabla\times F\vec n\,ds}$ where ${S'}$ is just the unit disk centered on the ${z}$ -axis sitting at ${z=2}$ . (This is also a consequence of the divergence theorem.) Conversely, Stoke’s can be use to calculate a line integral ${\oint_C F\cdot dr}$ via a corresponding surface integral ${\iint_S \nabla \times F\cdot\vec n\,d\sigma}$ where ${\partial S=C}$ .
The divergence theorem is just the three dimensional version of the normal form of Green’s theorem.
One important case (just as with Green’s and Stoke’s) is that when ${\nabla\cdot F=0}$ ( ${0}$ -divergence), then the flux integral through a surface is “independent of the surface” in some sense.
You should know the relevance of fields satisfying ${F(\vec r)=\frac{k}{|\vec r|^n}\hat r}$ (radial fields that are inversely proportional to the ${n^{\text{th}}}$ power of the distance from the origin and in particular that the divergence for these fields is ${0}$ when ${n=2}$ ).

Not to under emphasise this: Know the main theorems and definitions, know the conditions under which the theorems apply – i.e. the hypotheses of the theorems.

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This entry was posted on May 10, 2009 at 7:56 pm and is filed under Math 275, Spring 2009. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

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Math 275 – May 10 (Final Notes)

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