Mathematical Notation Math 221 - alculus & Analytic Geometry III Use Word or WordPerect to recreate the ollowing documents. Each article is worth 10 points and should be emailed to the instructor at james@richland.edu. I you use Microsot Works to create the documents, then you must print it out and give it to the instructor as he can't open those iles. Type your name at the top o each document. Include the title as part o what you type. The lines around the title aren't that important, but i you will type ----- at the beginning o a line and hit enter, both Word and WordPerect will draw a line across the page or you. For expressions or equations, you should use the equation editor in Word or WordPerect. The instructor used WordPerect and a 14 pt Times New oman ont with 0.75" margins, so they may not look exactly the same as your document. The equations were created using 14 pt ont. For individual symbols (θ, Φ, etc) within the text o a sentence, you can insert symbols. In Word, use "Insert / Symbol" and choose the Symbol ont. For WordPerect, use trl-w and choose the Greek set. However, it's oten easier to just use the equation editor as expressions are usually more complex than just a single symbol. I there is an equation, put both sides o the equation into the same equation editor box instead o creating two objects. There are instructions on how to use the equation editor in a separate document or on the website. Be sure to read through the help it provides. There are some examples at the end that walk students through the more diicult problems. You will want to read the handout on using the equation editor i you have not used this sotware beore. I you ail to type your name on the page, you will lose 1 point. Don't type the grayed out hints or reminders at the bottom o each page. These notations are due at the beginning o class on the day o the exam or that chapter. That is, the chapter 12 notation is due on the day o the chapter 12 test. Late work will be accepted but will lose 20% o its value per class period.
hapter 12 - Vectors Press trl-b beore each vector to make it bold Dot Products u v uv + uv + uv The dot product is a scalar and is deined as 1 1 2 2 3 3 The dot product is 0 i and only i the vectors are orthogonal. u v u v cosθ The orthogonal projection o v onto b is v b projbv b 2 b W F cosθ PQ F PQ Work can be ound by ross Products i j k u v u u u v v v The cross product is a vector and is deined as 1 2 3 The cross product is orthogonal to both vectors. u v u v sinθ 1 2 3 The area o a parallelogram is equal to the cross product o the adjacent side vectors. The cross product is the zero vector i and only i the vectors are parallel. Triple Scalar Products The triple scalar product is a scalar and is deined as u ( v w) u1 u2 u3 v1 v2 v3 w w w The absolute value o the triple scalar product is the volume o a parallelpiped. The triple scalar product is 0 i and only i the vectors are coplanar. 1 2 3 Don't orget to put your name at the top o the page
hapter 13 - Vector Valued Functions The derivatives o dot and cross products ollow the product rule. ( r1 r2) r 1 r2 + r1 r 2 and I a vector valued unction has constant length, the r and r' are orthogonal. The arc length o a smooth vector valued unction is d r d r dt The chain rule is dτ dt dτ The arc length parametrization is For a smooth vector valued unction, r r r r + r r 1 2 1 2 1 2 b d L r dt a Hold down the shit while selecting the I symbol to get it to grow with the integrand. d s r du t t 0 dr ds dt dt du and dr 1 ds To ind the unit tangent vector, take the derivative and normalize it. T() t dt r r To ind the unit normal vector, take the derivative o the unit tangent vector and normalize it. N T T For curves parametrized by arc length, T s r s and N( s) r r ( s) ( s) The binormal vector is the cross product o the unit tangent and unit normal vectors. It is also a unit ( t) B T N vector and is ound by. The curvature is deined by dt κ ds ( s) r s or κ The radius o the oscillating circle is called the radius o curvature and is T r ( t) 1 ρ κ Don't orget to put your name at the top o the page
hapter 14 - Partial Derivatives The irst-order partial derivative o with respect to x is denoted by x ( x, x and is ound by inding the derivative o with every variable other than x treated as a constant. The second-order partial derivatives o are xy, yx y x 2 2 xy xx x y, 2 2 2 yy 2 and the mixed partials are. I is continuous, the. Notice the ordering on the partials. The order or is rom let to right, x irst and y second. The order or is right to let, y irst and x second. xy yx 2 (, ) ( x0, y 0) ( x0, y0) (, ) (, ) z x y I is dierentiable at, then the total dierential o at is dz x y dx + x y dy x The chain rule says that i 0 0 y 0 0 dz dx dy + dt dt dt z z z + u u u z ( x, and x and y are both unctions o t, then. Furthermore, i x and y are both unctions o u and v, then and The gradient o is a vector deined by x, y, z xi + yj + zk z z z + v v v x y +, xi y. is read "del ". The directional derivative o in the direction o the unit vector u can be written as (, ) (, ) D x y x y u u 0 0 0 0 The applications o the gradient are too numerous to it on this page. It will appear in many ormulas. j or Don't orget to put your name at the top o the page
hapter 15 - Multiple Integrals To evaluate a deinite integral, work rom inside to outside. The order o the integration is important, so be sure to use proper notation. b g2( x) d h2( 1 1 x, y da x, y dy dx x, y dx dy a g x c h y I you can split the integrand into two independent unctions o x and y and the limits are constant, then you can split a double (or triple) integral up into the product o the integrals. b d b d x g y dy dx x dx g y dy a c a c The area o a region is A da. The volume o a solid G is The surace area o a parametrically deined surace σ is S V G r r u v y ( x, y ) δ (, ) The center o gravity o a lamina is given by, M x 1 y yδ ( x, da M mass o dv da M 1 x x x y da M mass o The Theorem o Pappus says that the volume o a solid by revolving a region about a line L is the area o the region times the distance traveled by the centroid. For polar and cylindrical coordinates, you need to insert an extra r into the integrand. For spherical coordinates, you need to insert an extra ρ 2 sinφ into the integrand. All o this is related to the Jacobian. I T is a transormation rom the uv plane into the xy plane, then the Jacobian o T is denoted by (, ) J u v ( xy, ) u u ( uv, ) x y v v by the absolute value o the Jacobian.. When applying the transormation, multiply the integrand Don't orget to put your name at the top o the page
hapter 16 - Topics in Vector alculus ( x, y, z) ( x, y, z) + g ( x, y, z) + h( x, y, z) F i j k g h F + + F z onsider the vector unction. The divergence o F is a scalar deined by div The curl o F is a vector deined by curl i j k F F z g h The line integral o with respect to s along is the net signed area between the curve and the graph ( x, y ) (, ) A x y ds o and is denoted by. L Arc length can be expressed as. ds The value o a line integral does not depend on its parametrization. However, i the orientation is reversed, the sign o the integral with respect to x and y changes, but the integral with respect to the arc length parameter s remains unchanged. The work perormed by the vector ield is W F T ds F d r I F is a conservative vector ield and F( x, φ( x, calculus applies to line integrals and ( x, d r φ( x, y ) φ( x, y ) F then the irst undamental theorem o 1 1 2 2 I F is a conservative vector ield, then the integral is independent o the path. F d r 0 or every piecewise smooth closed curve and A vector ield in 2-space is conservative i Green's Theorem says (, ) (, ) g and in 3-space i curl F 0 g x y dx + g x y dy da Don't orget to put your name at the top o the page