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<title>
Test 3 - review
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<h1>Test 3</h1>
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<p class="show">Bring
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<li>calculator
<li>one 8 1/2 X 11 page of notes that you prepare ahead of time.
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<p>Of necessity, the test will be <b>open book</b>.  You may use a calculator (make sure you know how to switch between degrees and radians!). You may consult your notes, my notes, the textbook, use a calculator, use CoCalc,*but not other people*.

<p>The test will be timed.  I'll send out a .pdf of the exam at 12:00 PM--noon (Goshen time) and end at 2:00 PM.  You may mark it up (Notability) or write your answers on separate pieces of paper, and upload snapshots of them in when finished.


<h3  class="show">Topics</h3>

<h4>Double Integrals</h4> 

<p>Be able to 
<ul>
<li>set up a double integral over an arbitrary area
<li>do an integration in terms of the 
volume of a solid above an arbitrary area.

<li>compute double integrals. 
<li>to determine limits of integration and change the order of
integration. 
<li>Know what dA means and what dA is in rectangular and in
polar coordinates.
</ul>


<h4>Triple Integrals</h4>

<p> Be able to 
<ul><li>compute triple integrals. 
<li>set up triple
integrals to compute the volume of a solid. 
<li>set up and
compute triple integrals of a function of 3 variables over a solid
region (Cartesian, cylindrical, and spherical). 
<li>change the order of integration and modify the
limits appropriately. 
<li>Know what dV means and what dV is in
rectangular, cylindrical, and spherical coordinates.
</ul>

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<h4>Coordinate transformations</h4>

<p>Be able to 
<ul><li>verify and use a coordinate transformation:  $x(u,v,w), y(u,v,w), z(u,v,w)$; Check that it maps $(x,y,z)\mapsto (u,v,w)$ correctly.
<li>calculate the Jacobian for a coordinate transformation, given the coordinate mapping, and know the interpretation of the Jacobian as the ratio of areas in the two spaces.
</ul>
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<h4>Vector Fields</h4>

<p> Be able to 
<ul><li>identify and draw vector fields. 
<li>test if a
vector field is conservative, and, if so... 
<li>to find a
potential function associated with that vector field.
<li>Understand the
relationship between gradient fields and level curves. 
<li>Know other consequences if
a vector field to be conservative.  (E.g. irrotational, path independence...)
</ul>

<h4>Curl and Divergence</h4>

<p> Be able to 
<ul><li>compute the curl and divergence of a vector
field. 
<li>do a
graphical/visual analysis of a vector field, to find the sign of curl and
divergence. 
<li>Understand what it means for a vector field to be
irrotational or incompressible and be able to identify whether or not
a vector field is irrotational or incompressible.
</ul>
<h4>Line Integrals</h4>

<p> Be able to 
<li>compute the line integral of a scalar function over a
curve, including parameterizing the curve if necessary. 
<li>compute the line integral of a vector function (either the tangential component or the normal component) over
a curve.
<li>interpret "work that a field does on a particle" as the tangential component of a force field, $\myv F\cdot\uv T$, integrated over the trajectory of a particle.
<li>estimate the sign of a vector line integral, based on the trajectory and a sketch of nearby vector field arrows.
<li>Understand when a line integral is path independent and the
relationships between path independence, gradient fields, conservative
vector fields.

</ul>

<h4>Green's Theorem</h4>

<p>Be able to 
<ul><li>Know the statement of Green's Theorem (both forms: in terms of tangential and normal components of vector fields). 
<li>apply Green's Theorem to convert line integrals $\leftrightarrow$ area integrals.
<li>(* nope, not this time:) to use the divergence theorem to convert a surface flux integral (3-d) to a volume integral of divergence.  
</ul>


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