http://sites.science.oregonstate.edu/BridgeBook/ 2020-01-26T16:02:24-08:00 http://sites.science.oregonstate.edu/BridgeBook/ http://sites.science.oregonstate.edu/BridgeBook/lib/images/favicon.ico text/html 2012-09-27T02:56:00-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/sguidecontent/changes?rev=1348739760 The vector from the origin to the point $(x,y,z)$ is $$\rr=x\,\ii+y\,\jj+z\,\kk$$ and is called the position vector. A parametric curve can be described by viewing $x$, $y$, and $z$, and hence $\rr$, as functions of some parameter $u$. It is instructive to draw a picture showing the small change $\Delta\rr=\Delta x\,\ii+\Delta y\,\jj+\Delta z\,\kk$ in the position vector between nearby points. Try it! text/html 2012-09-26T20:38:00-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/sguidecontent/cov?rev=1348717080 RECALL: $dA=dx\,dy=r\,dr\,d\phi$ Where did the factor of $r$ come from in the above expression? Consider a ``coordinate rectangle'' bounded by curves of the form $r=\hbox{constant}$, $\phi=\hbox{constant}$. If this ``rectangle'' is small enough, its sides have length $dr$ and $r\,d\phi$, so that its area is the product $(dr)(r\,d\phi)$. text/html 2012-09-26T20:29:00-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/sguidecontent/cross?rev=1348716540 Cross Product The cross product is fundamentally a directed area. The magnitude of the cross product is defined to be the area of the parallelogram, as shown in the figure above. This leads to the formula $$ |\vv\times\ww| = |\vv||\ww|\sin\theta $$ an immediate consequence of which is that $$ \vv\parallel\ww \Longleftrightarrow \vv\times\ww=\zero $$ The direction of the cross product is given by the right-hand rule, so that in the example shown $\vv\times\ww$ points out of the page. This i… text/html 2015-08-27T18:04:00-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/sguidecontent/curl?rev=1440723840 RECALL: $\grad{f} = {\partial f\over\partial x}\,\ii + {\partial f\over\partial y}\,\jj + {\partial f\over\partial z}\,\kk$ It is convenient to think of $\grad$ as a differential operator (called ``del''), written as $$\grad = {\partial\over\partial x}\,\ii + {\partial\over\partial y}\,\jj + {\partial\over\partial z}\,\kk$$ Thus, $\grad{f}$ (``del f'') is obtained by ``multiplying'' $\grad$ by $f$, being careful of course to interpret ``multiplication'' as partial differentiation. text/html 2012-09-26T16:09:00-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/sguidecontent/differentials?rev=1348700940 Given a function $f=f(x)$, the differential of $f$ is defined to be $$df = {df\over dx} \,dx$$ This definition is not very satisfying. What, after all, is $dx$? There are several ways to make this precise. In the branch of mathematics known as differential geometry, $dx$ and $df$ are differential forms, but this is more sophistication than we need. text/html 2012-09-26T21:29:00-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/sguidecontent/divergence?rev=1348720140 At any point $P$, we define the divergence of a vector field $\FF$, which we write for now as $\Div\FF$, to be the flux of $\FF$ per unit volume ``at'' $P$, which is the (limiting value of the) flux per unit volume out of a small box around $P$. Furthermore, we can chop up an arbitrary (closed) box into a suitable combination of (small) rectangular boxes. The outward fluxes through all interior sides will cancel, leaving just the flux out of the original box. Thus, \begin{equation} \Int_{\rm b… text/html 2012-09-26T21:40:00-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/sguidecontent/fields?rev=1348720800 It is time to distinguish between several different vector-like objects. The arrow pointing from the origin to the point with (Cartesian) coordinates $(a,b,c)$ is $$\ww = a\,\ii + b\,\jj + c\,\kk$$ This is a vector, which is said to have its tail at the origin, or to live at the origin. There is nothing special about the origin; vectors can live at any point. text/html 2015-08-27T18:04:00-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/sguidecontent/green?rev=1440723840 The circulation of a vector field $\FF$ around a closed loop $C$ is simply the integral $$\Oint \FF\cdot d\rr$$ We will restrict ourselves in this lesson to the $xy$-plane; the circle on the integral sign then implies both that $C$ is closed, and that the loop is oriented in a counterclockwise direction. text/html 2012-09-26T21:26:00-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/sguidecontent/integral?rev=1348719960 The main results of the previous section were formulas for the surface elements $d\SS$ and $\dS$ of a surface in terms of two infinitesimal displacements in the surface, that is $$d\SS = d\rr_1 \times d\rr_2$$ and $$\dS = |d\SS| = |d\rr_1\times d\rr_2|$$ which are~(1) and (2) of Section 8, respectively. This allows us to integrate any function over a surface, by considering an integral of the form $$\Sint f \, \dS$$ The surface area of $S$ is obtained as a special case by setting $f=1$. Anoth… text/html 2012-09-25T20:59:00-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/sguidecontent/intro?rev=1348631940 This is a Study Guide for MTH 255, Vector Calculus II. The Study Guide can in principle be used with any calculus text, although it was developed using both the fourth edition of Calculus by James Stewart (either the Multivariable or Early Transcendentals versions) and the third edition of Multivariable Calculus by William McCallum, Deborah Hughes-Hallett, Andrew Gleason, et al. A rough correspondence between sections in (more recent editions of) those texts and lessons in the Study Guide ca… text/html 2012-09-26T16:54:00-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/sguidecontent/line?rev=1348703640 If you want to add up something along a curve, you need to compute a line integral. Common examples are the length of a curve, how much a wire weighs, and how much work is done when moving along a particular path. We start with the last of these, namely finding the work $W$ done by a force $\FF$ in moving a particle along a curve $C$. We begin with the relationship $$\hbox{work} = \hbox{force} \times \hbox{distance}$$ Suppose you take a small step $d\rr$ along the curve. How much work was do… text/html 2015-08-27T18:04:00-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/sguidecontent/master?rev=1440723840 Take another look at the formula for the differential of a function of several variables: $$ df = \Partial{f}{x}\,dx + \Partial{f}{y}\,dy + \Partial{f}{z}\,dz $$ Each term is a product of two factors, labeled by $x$, $y$, and $z$. This looks like a dot product! Separating out the pieces, we have $$ df = \left( \Partial{f}{x}\,\ii + \Partial{f}{y}\,\jj + \Partial{f}{z}\,\kk \right) \cdot (dx\,\ii + dy\,\jj + dz\,\kk) $$ The last factor is just $d\rr$, and you may recognize the first factor a… text/html 2012-09-26T21:36:00-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/sguidecontent/mmm?rev=1348720560 We describe here a variation of the usual procedure for determining whether a vector field is conservative and, if it is, for finding a potential function. It is helpful to make a diagram of the structure underlying potential functions and conservative vector fields. For functions of two variables, this is shown in the first diagram above. The potential function $f$ is shown at the top. Slanted lines represent derivatives of $f$; derivatives with respect to $x$ go to the left, while derivati… text/html 2012-09-25T20:42:00-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/sguidecontent/notation?rev=1348630920 WARNING: This Study Guide does NOT use the same conventions for spherical coordinates as those used in your text. Instead, the Study Guide uses the most commonly used choice in science and engineering applications, explicitly given by $$x=r\sin\theta\cos\phi \qquad y=r\sin\theta\sin\phi \qquad z=r\cos\theta$$ You will often see $\rho$ (or $R$) instead of $r$, and/or the roles of $\theta$ and $\phi$ interchanged. However, spherical coordinates should ALWAYS be given in the following order: $$\hb… text/html 2015-08-27T18:04:00-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/sguidecontent/path?rev=1440723840 RECALL: $\displaystyle\Int_a^b{df\over dx}\,dx = f \Big|_a^b$ This is the Fundamental Theorem of Calculus, which just says that the integral of a derivative is the function you started with. We could also write this simply as $$\int df = f$$ But recall the master formula~(1) of Section 3 which says $$df = \grad{f} \cdot d\rr$$ Putting this all together, we get the fundamental theorem for line integrals, which says that $$\Lint \grad{f} \cdot d\rr = f \Big|_A^B$$ for any curve $C$ starting … text/html 2012-09-25T20:43:00-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/sguidecontent/review?rev=1348630980 Most texts contain a very nice summary of the main theorems. Here is a partial list of topics you should review: \begin{itemize}\item The Gradient: You should understand what the gradient of a function means geometrically and physically. (Can you relate the gradient to the level sets of the function?) \item Line Integrals: You should be able to compute line integrals, determine whether vector fields are conservative, be able to find potential functions, and know the Fundamental Theorem for Li… text/html 2012-09-26T21:31:00-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/sguidecontent/stokes?rev=1348720260 At any point $P$, we define the curl of a vector field $\FF$, which we write for now as $\Curl\FF$, to be the vector which gives the circulation of $\FF$ per unit area ``at'' $P$. That is, given any unit vector $\nn$ at $P$, $(\Curl\FF)\cdot\nn$ should be the (limiting value of the) circulation of $\FF$ per unit area at $P$ around a small loop with axis $\nn$. text/html 2012-09-26T21:48:00-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/sguidecontent/surface?rev=1348721280 There are many ways to describe a surface. Consider the following following descriptions: \begin{itemize}\item the unit sphere; \item $x^2+y^2+z^2=1$; \item $r=1$ (where $r$ is the spherical radial coordinate); \item $x=\sin\theta\cos\phi$, $y=\sin\theta\sin\phi$, $z=\cos\theta$, with $\theta\in[0,\pi]$ and $\phi\in[0,2\pi)$; \item $\rr = x\,\ii + y\,\jj + z\,\kk$, with $x$, $y$, $z$ given as above; \end{itemize} text/html 2012-09-25T20:48:00-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/sguidecontent/vectors?rev=1348631280 Vectors A vector $\ww$ is an arrow in space, having both a magnitude and a direction. Examples of vectors include the displacement from one point to another, and your velocity when moving along some path. The magnitude of $\ww$ is denoted by $|\ww|$, also written $||\ww||$. This is often casually called the ``length'' of $\ww$, but that is only technically correct if $\ww$ has units of length. The displacement between two points does indeed have a length, but to talk about your speed as bei…