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Y x A B 130º (a) Angle between y and A 90 50 140 ( , ) ( ) 90 ( ) ,( ) ˆ , ˆ plane A B xy b Angle y A B C angle j k because C perpendicular P4 Vectors A and B lie in an xy plane A has a magnitude 800 and angle 130º;`B `C `D B D`J C We get to choose , , and , so there are several posJ J JB C D sible vector fields with a given divergence (This is similar to the freedom enjoyed when finding a vector field with a given rotation) (a) works, as does , , and so forthF i F j F kœB œC œD (b) Three possible solutions are , , and F i F j F kœ B C œ B C œBCurves in R2 Three descriptions (1) Graph of a function f R !R (That is y= f(x)) Such curves must pass the vertical line test Example When we talk about the \curve" y= x2, we actually mean to say the graph of the function f(x) = x2That is, we mean the set

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Line ƒAƒCƒRƒ" ‰Âˆ¤‚¢ ƒfƒBƒYƒj[-Solution to Problem Set #4 1 (a) (15 pts) Find parametric equations for the tangent line to the curve r(t) = ht3,5t,t4i at the point (−1,−5,1) (b) (15 pts) At what point on the curve r(t) = ht3,5t,t4i is the normal plane (this is the plane that is perpendicular to the tangent lineY)~j ·~r where C is the curve y = sin(x) from (0,0) to (π,0) and the line segment from (π,0) to (0,0) First observe that this curve is oriented in the wrong direction (see illustration) However, as we observed before, we have R −C F~ d~r = − R C F~ d~r, so we need to find the line integral and then simply negate the answer

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A 1 B 115 C 65 D 60 1º a b xº c d 115º VA_GM_RB 3/4/11 735 AM Page 5F G 24 H 25 J 33 (3 x – 35)° m (7 x – 115)° r 3 If lines a and b are parallel, what is the value of x?Evaluate the line integral Z C F~d~r, where C is the curve described by x2 y2 = 9 and z= 4, oriented clockwise when viewed from above Now, we just need to evaluate the line integral, using the de nition of the line integral (This is like #4(a) on the worksheet \Vector Fields and Line Integrals") We start by parameterizing C

C F·dr = Z C ∇f ·dr = f(r(1))− f(r(0)) = f(0,1,1)− f(1,0,0) = −1 −0 = −1 Problem 3 Use Green's theorem to evaluate the line integral I C (x 3− y ) dx(x3 y3) dy where C is the oriented curve shown in Figure 1 x y (−2,0) (−1,0) (1,0) (2,0) Figure 1 C is the union of two semicircles and two line segments Solution CSurface area and surface integrals (Sect 165) I Review Arc length and line integrals I Review Double integral of a scalar function I The area of a surface in space Review Double integral of a scalar function I The double integral of a function f R ⊂ R2 → R on a region R ⊂ R2, which is the volume under the graph of f and above the z = 0 plane, and is given byAnd cost = y=3 So sin2 tcos2 t = x2 4 y2 9 = 1 The curve is an ellipse with center at (0;0) and the major axis along the yaxis x ( /3) (p

Background The efficacy of the ALK inhibitor crizotinib as compared with standard chemotherapy as firstline treatment for advanced ALKpositive nonsmallcell lung cancer (NSCLC) is unknown Methods We conducted an openlabel, phase 3 trial comparing crizotinib with chemotherapy in 343 patients with advanced ALKpositive nonsquamous NSCLC who had received no previousA line is a collection of points along a straight path with no end points A line segment is a part of a line that contains every point on the line between its end points B C N M X Y S R J K L E F Plane N LM PQ N P Q Points, Lines and Planes Score Printable Math Worksheets @ wwwmathworksheets4kidscom Name Sheet 3 Printable MathAbcdefg hijklmn opqrstu vwxyz(ed) French Canadian version A Frenchlanguage version of the song is also taught in Canada, with generally no alterations to the melody except in the final line that requires adjustment to accommodate the twosyllable pronunciation of the French y Phonics songs

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Parallel lines e and f are cut by transversal b Horizontal and parallel lines e and f are cut by transversal b At the intersection of lines b and e, the uppercase right angle is (2 x 18) degrees At the intersection of lines b and f, the top right angle is (4 x minus 14) degrees and the bottom right angle is y degrees What is the value of y?B has components B x=772,B y= 9 What areLine Integrals and Green's Theorem Jeremy Orlo 1 Vector Fields (or vector valued functions) Vector notation In 1804 we will mostly use the notation (v) = (a;b) for vectors

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Assignment 6 (MATH 215, Q1) 1 Evaluate the line integral with respect to arc length (a) Z C xey ds where C is the arc of the unit circle from (1,0) to (−1,0) traversed counterclockwise Solution C can be parametrized asDepartment of Mechanical Engineering Characteristics of forces Force Vector with magnitude and direction Magnitude – a positive numerical value representing the size or amount of the force Directions – the slope and the sense of a line segment used to represent the force – Described by angles or dimensions2 Find the measure of x in quadrilateral A ⁢ B ⁢ G ⁢ C Quadrilateral with one angle labelled 7x3, one angle 6x5, one angle 16x 15, and one angle 4x 23 For problems 3–7, use the graph of lines p

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 Section 52 Line Integrals Part I In this section we are now going to introduce a new kind of integral However, before we do that it is important to note that you will need to remember how to parameterize equations, or put another way, you will need to be able to write down a set of parametric equations for a given curveHere, y= f(x) f(a), while dy= f0(a) x= f0(a)(x a) So f(x) f(a) ˇf0(a)(x a) Therefore f(x) ˇf(a) f0(a)(x a) The righthand side f(a) f0(a)(x a) can be interpreted as follows It is the best linear approximation to f(x) at x= a It is the 1st Taylor polynomial to f(x) at x= a The line y= f(a) f0(a)(x a) is the tangent line at (a;f(a))C×r = 1 i j k c c2 c3 x y z 2 = (c z −c 3y,c x −c 1z,c y −c2x) (i) Using the identity div(fF) = f divFgrad(f) ·F, and setting f = rn and F = c×r gives div(rn(c×r)) = rn div(c×r) grad(rn)(c×r) = 0nrn−2r(c×r) = 0 This uses the result from Example 45, and the fact that c×r) is perpendicular to r (ii), using the

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The addition of atezolizumab to bevacizumab plus chemotherapy significantly improved progressionfree survival and overall survival among patients with metastatic nonsquamous NSCLC, regardless of PDL1 expression and EGFR or ALK genetic alteration status (Funded by F HoffmannLa Roche/Genentech;F B F C 075 ft 30 B C 25 Determine the moment of each force about the bolt located at ATake F B =40 lb, C 50 lb SOLUTION ad Ans adM C = 50 cos 30°(325) = 141 lb # ft Ans M B = 40 cos 25°(25) = 906 lb # ft27 Tangent Planes to Level Surfaces Suppose S is a surface with equation F(x, y, z) = k, that is, it is a level surface of a function F of three variables, and let P(x 0, y 0, z 0) be a point on S Let C be any curve that lies on the surface S and passes through the point PRecall that the curve C is described by a continuous vector function r(t) = 〈x(t), y(t), z(t)〉

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This page lists the letters of the English alphabet from a to z Vocabulary for ESL learners and teachersIf x, y, z∈ Rn, we have h x y z i ∈ Rn×3, a matrix with columns x, y, and z We can construct a block vector as (x,y,z) = x y z ∈ R3n Functions The notation f A→ Bmeans that fis a function on the set Ainto the set B The notation b = f(a) means b is the value of the function f at the point a, where a ∈ A and b ∈ BThis integral of a function along a curve C is often written in abbreviated form as ∫ C f ( x, y) d s Example 1621 Compute ∫ C y e x d s where C is the line segment from ( 1, 2) to ( 4, 7) We write the line segment as a vector function r = 1, 2 t 3, 5 , 0 ≤ t ≤ 1, or in parametric form x = 1 3 t, y

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 Section 54 Line Integrals of Vector Fields In the previous two sections we looked at line integrals of functions In this section we are going to evaluate line integrals of vector fields We'll start with the vector field, →F (x,y,z) =P (x,y,z)→i Q(x,y,z)→j R(x,y,z)→k F → ( x, y, z) = P ( x, y, z) i → Q ( x, y, z) j111 Functions and straight lines A Functions 1 Deflnltlon Afunctlon defined on a certain set of real numbers D (called the domain of the function) is a(b) The paramentric equation of C is (x = 2cost y = 2sint, 0 ≤ t ≤ 2π Thus Z C f ds = Z 2π 0 4cos2 t−4sin2 t p 4sin2 t4cos2 tdt = 8 Z 2π 0 cos(2t) = 4sin(2t) 2π 0 = 0 3 Evaluate Z C (xyz)dx2xdyxyzdz, where C consists of line segments from (1,0,1) to (2,3,1)

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Find stepbystep Calculus solutions and your answer to the following textbook question Evaluate the line integral C Fdr where C is given by the vector function r(t)F(x,y,z)=sinxicosyjxzk, r(t)=t^3it^2jtk, 0Search the world's information, including webpages, images, videos and more Google has many special features to help you find exactly what you're looking forLn(b/a) µT/W C pε ln(D/a) 2pε ln(b/a) εW/T In this table, D and a are the centertocenter spacing and wire radius of the twowire line, b and a are the outer and inner radius of the coaxial line and T and W are the dielectric thickness and conductor width of the microstrip line For twowire line, the expressions include the

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B Line m and line p c Q and R are planes m p Œ F R E D EXAMPLE 1 Name Points, Lines, and Planes Use the diagram at the right a Name three points that are collinear b Name four points that are coplanar c Name three points that are not collinear Solution a Points D, E, and F lie on the same line So, they are collinearLine, surface and volume integrals, curvilinear coordinates 5 Vector operators — grad, div and curl 6 Vector Identities, curvilinear coordinate systems dV = (∇∇ ·f) dV dS = dS = j z x y dz dx dy −dxdzj dxdz • Conclusion The divergence of a vector field represents the flux generation per unit volume at each point of thePass (alphabetical) M Arehdall, E H Bottrell, E A Brearley, G A Brookes, F W D Collier, A E Colvin, Daisy E Culpin, A W Fox, R I Furber, C B

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From this geometry, we can see that line integral ∫ C f (x, y) d s ∫ C f (x, y) d s does not depend on the parameterization r (t) r (t) of C As long as the curve is traversed exactly once by the parameterization, the area of the sheet formed by the function and the curve is the same This same kind of geometric argument can be extended toWe use the extended form of Green's theorem to show that ∮ C F d r ∮ C F d r is either 0 or −2 π −2 π —that is, no matter how crazy curve C is, the line integral of F along C can have only one of two possible values We consider two cases the case when C encompasses the origin and the case when C does not encompass the origin Case 1 C Does Not Encompass the Origin163 The Fundamental Theorem of Line Integrals One way to write the Fundamental Theorem of Calculus ( 721) is ∫ a b f ′ ( x) d x = f ( b) − f ( a) That is, to compute the integral of a derivative f ′ we need only compute the values of f at the endpoints Something similar is true for line

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131 Metric spaces 275 Example 1312 For R2 with the Euclidean metric de ned in Example 136, the ball B r(x) is an open disc of radius rcentered at xFor the '1metric in Exam ple 135, the ball is a diamond of diameter 2r, and for the '1metric in Exam ple 137, it is a square of side 2rGiven two points a,b lying on a surface S ⊂ R3, we seek the curve C ⊂ S that joins them and has the minimal possible length For example, if S is a circular cylinder, then there are three possible types of geodesic curves straight line segments parallel to the center lineC f(x;y)ds = Z b a f(x(t);y(t))j!r0(t)jdt (53) = Z b a f(x(t);y(t)) s dx dt 2 dy dt dt Remark 391 We used aand bfor the limits of integration because they are the limits of the variable t Remark 392 Note that the line integral is with respect to arc length However, to compute it, we use the parametrization of the curve, whatever it is We

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(ii) by using the Fundamental Theorem for line integrals 4C3 Let f(x,y) = sinxcosy a) Calculate F = ∇f b) What is the maximum value R C F dr can have over all possible paths Cin the plane?A) Calculate F = ∇f b) Calculate the integral R C F·dr (i) directly;B D G C F E q p r SOLUTION 1 Understand the Problem In the diagram, you are given three lines, p, q, and r, that intersect at point B You need to name two different planes that contain line r 2 Make a Plan The planes should contain two points on line r and one point not on line r 3 Solve the Problem Points D and F are on line r Point E

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Give a path Cfor which this maximum value is attainedT line Determine equations of geo desics on the follo wing surfaces a Righ t circular cylinder T ak e ds t d dz and minimi ze Z v u u dz d or Z v u u t a d dz dz b J y ZZ R K s t s t dsdt Z b a dt f where K s t is a giv en con tin uous function of s and t on the square R for whic h a b K s t is symmetri c and f t is con tin uous Hin t the answ8 (a) Sketch the plane curve r(t) = (2sint)i(3cost)j (b) Find r0(t) (c) Sketch the position vector r(t) and the tangent vector r0(t) for t = ˇ=3 Solution (a) x = 2sint, y = 3cost =) sint = x=2;

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Here r is the position vector from point O to any point on the line of action of F MOMENT OF A FORCE – VECTOR FORMULATION (continued) So, using the cross product, a r x F B) F x r C) r • F D) r * F CONCEPT QUIZ 1 If a force of magnitude F can be applied in four different 2D254 CHAPTER 13 CALCULUS OF VECTORVALUED FUNCTIONS (LT CHAPTER 14) Use a computer algebra system to plot the projections onto the xy and xzplanes of the curve r(t) = t cost,tsin t,t in Exercise 17 In Exercises 19 and , let r(t) = sin t,cost,sin t cos2t as shown in Figure 12 y x z FIGURE 12 19Of f(x) and f(y) Geometrically, the line segment connecting (x;f(x)) to (y;f(y)) must sit above the graph of f If f is continuous, then to ensure convexity it is enough to check the de nition with = 1 2 (or any other xed 2(0;1)) This is similar to the notion of midpoint convex sets that we saw earlier We say that fis concave if fis convex

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144 8 Di erentiable Functions is approximated near cby the linear function h7!f0(c)h Thus, f0(c) may be interpreted as a scaling factor by which a di erentiable function fshrinks or stretches lengths near c If jf0(c)j1, then f stretches the length of an interval5 2 Lines m and r are cut by a transversal What value of x will show that line m is parallel to line r ?

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