Approaching Green's Theorem Via Riemann Sums
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The Directional Derivative the Derivative of a Real Valued Function (Scalar field) with Respect to a Vector
Math 253 The Directional Derivative The derivative of a real valued function (scalar field) with respect to a vector. f(x + h) f(x) What is the vector space analog to the usual derivative in one variable? f 0(x) = lim − ? h 0 h ! x -2 0 2 5 Suppose f is a real valued function (a mapping f : Rn R). ! (e.g. f(x; y) = x2 y2). − 0 Unlike the case with plane figures, functions grow in a variety of z ways at each point on a surface (or n{dimensional structure). We'll be interested in examining the way (f) changes as we move -5 from a point X~ to a nearby point in a particular direction. -2 0 y 2 Figure 1: f(x; y) = x2 y2 − The Directional Derivative x -2 0 2 5 We'll be interested in examining the way (f) changes as we move ~ in a particular direction 0 from a point X to a nearby point . z -5 -2 0 y 2 Figure 2: f(x; y) = x2 y2 − y If we give the direction by using a second vector, say ~u, then for →u X→ + h any real number h, the vector X~ + h~u represents a change in X→ position from X~ along a line through X~ parallel to ~u. →u →u h x Figure 3: Change in position along line parallel to ~u The Directional Derivative x -2 0 2 5 We'll be interested in examining the way (f) changes as we move ~ in a particular direction 0 from a point X to a nearby point . -
Cauchy Integral Theorem
Cauchy’s Theorems I Ang M.S. Augustin-Louis Cauchy October 26, 2012 1789 – 1857 References Murray R. Spiegel Complex V ariables with introduction to conformal mapping and its applications Dennis G. Zill , P. D. Shanahan A F irst Course in Complex Analysis with Applications J. H. Mathews , R. W. Howell Complex Analysis F or Mathematics and Engineerng 1 Summary • Cauchy Integral Theorem Let f be analytic in a simply connected domain D. If C is a simple closed contour that lies in D , and there is no singular point inside the contour, then ˆ f (z) dz = 0 C • Cauchy Integral Formula (For simple pole) If there is a singular point z0 inside the contour, then ˛ f(z) dz = 2πj f(z0) z − z0 • Generalized Cauchy Integral Formula (For pole with any order) ˛ f(z) 2πj (n−1) n dz = f (z0) (z − z0) (n − 1)! • Cauchy Inequality M · n! f (n)(z ) ≤ 0 rn • Gauss Mean Value Theorem ˆ 2π ( ) 1 jθ f(z0) = f z0 + re dθ 2π 0 1 2 Related Mathematics Review 2.1 Stoke’s Theorem ¨ ˛ r × F¯ · dS = F¯ · dr Σ @Σ (The proof is skipped) ¯ Consider F = (FX ;FY ;FZ ) 0 1 x^ y^ z^ B @ @ @ C r × F¯ = det B C @ @x @y @z A FX FY FZ Let FZ = 0 0 1 x^ y^ z^ B @ @ @ C r × F¯ = det B C @ @x @y @z A FX FY 0 dS =ndS ^ , for dS = dxdy , n^ =z ^ . By z^ · z^ = 1 , consider z^ component only for r × F¯ 0 1 x^ y^ z^ ( ) B @ @ @ C @F @F r × F¯ = det B C = Y − X z^ @ @x @y @z A @x @y FX FY 0 i.e. -
1 Curl and Divergence
Sections 15.5-15.8: Divergence, Curl, Surface Integrals, Stokes' and Divergence Theorems Reeve Garrett 1 Curl and Divergence Definition 1.1 Let F = hf; g; hi be a differentiable vector field defined on a region D of R3. Then, the divergence of F on D is @ @ @ div F := r · F = ; ; · hf; g; hi = f + g + h ; @x @y @z x y z @ @ @ where r = h @x ; @y ; @z i is the del operator. If div F = 0, we say that F is source free. Note that these definitions (divergence and source free) completely agrees with their 2D analogues in 15.4. Theorem 1.2 Suppose that F is a radial vector field, i.e. if r = hx; y; zi, then for some real number p, r hx;y;zi 3−p F = jrjp = (x2+y2+z2)p=2 , then div F = jrjp . Theorem 1.3 Let F = hf; g; hi be a differentiable vector field defined on a region D of R3. Then, the curl of F on D is curl F := r × F = hhy − gz; fz − hx; gx − fyi: If curl F = 0, then we say F is irrotational. Note that gx − fy is the 2D curl as defined in section 15.4. Therefore, if we fix a point (a; b; c), [gx − fy](a; b; c), measures the rotation of F at the point (a; b; c) in the plane z = c. Similarly, [hy − gz](a; b; c) measures the rotation of F in the plane x = a at (a; b; c), and [fz − hx](a; b; c) measures the rotation of F in the plane y = b at the point (a; b; c). -
1 Mean Value Theorem 1 1.1 Applications of the Mean Value Theorem
Seunghee Ye Ma 8: Week 5 Oct 28 Week 5 Summary In Section 1, we go over the Mean Value Theorem and its applications. In Section 2, we will recap what we have covered so far this term. Topics Page 1 Mean Value Theorem 1 1.1 Applications of the Mean Value Theorem . .1 2 Midterm Review 5 2.1 Proof Techniques . .5 2.2 Sequences . .6 2.3 Series . .7 2.4 Continuity and Differentiability of Functions . .9 1 Mean Value Theorem The Mean Value Theorem is the following result: Theorem 1.1 (Mean Value Theorem). Let f be a continuous function on [a; b], which is differentiable on (a; b). Then, there exists some value c 2 (a; b) such that f(b) − f(a) f 0(c) = b − a Intuitively, the Mean Value Theorem is quite trivial. Say we want to drive to San Francisco, which is 380 miles from Caltech according to Google Map. If we start driving at 8am and arrive at 12pm, we know that we were driving over the speed limit at least once during the drive. This is exactly what the Mean Value Theorem tells us. Since the distance travelled is a continuous function of time, we know that there is a point in time when our speed was ≥ 380=4 >>> speed limit. As we can see from this example, the Mean Value Theorem is usually not a tough theorem to understand. The tricky thing is realizing when you should try to use it. Roughly speaking, we use the Mean Value Theorem when we want to turn the information about a function into information about its derivative, or vice-versa. -
MATH 25B Professor: Bena Tshishiku Michele Tienni Lowell House
MATH 25B Professor: Bena Tshishiku Michele Tienni Lowell House Cambridge, MA 02138 [email protected] Please note that these notes are not official and they are not to be considered as a sub- stitute for your notes. Specifically, you should not cite these notes in any of the work submitted for the class (problem sets, exams). The author does not guarantee the accu- racy of the content of this document. If you find a typo (which will probably happen) feel free to email me. Contents 1. January 225 1.1. Functions5 1.2. Limits6 2. January 247 2.1. Theorems about limits7 2.2. Continuity8 3. January 26 10 3.1. Continuity theorems 10 4. January 29 – Notes by Kim 12 4.1. Least Upper Bound property (the secret sauce of R) 12 4.2. Proof of the intermediate value theorem 13 4.3. Proof of the boundedness theorem 13 5. January 31 – Notes by Natalia 14 5.1. Generalizing the boundedness theorem 14 5.2. Subsets of Rn 14 5.3. Compactness 15 6. February 2 16 6.1. Onion ring theorem 16 6.2. Proof of Theorem 6.5 17 6.3. Compactness of closed rectangles 17 6.4. Further applications of compactness 17 7. February 5 19 7.1. Derivatives 19 7.2. Computing derivatives 20 8. February 7 22 8.1. Chain rule 22 Date: April 25, 2018. 1 8.2. Meaning of f 0 23 9. February 9 25 9.1. Polynomial approximation 25 9.2. Derivative magic wands 26 9.3. Taylor’s Theorem 27 9.4. -
The Mean Value Theorem) – Be Able to Precisely State the Mean Value Theorem and Rolle’S Theorem
Math 220 { Test 3 Information The test will be given during your lecture period on Wednesday (April 27, 2016). No books, notes, scratch paper, calculators or other electronic devices are allowed. Bring a Student ID. It may be helpful to look at • http://www.math.illinois.edu/~murphyrf/teaching/M220-S2016/ { Volumes worksheet, quizzes 8, 9, 10, 11 and 12, and Daily Assignments for a summary of each lecture • https://compass2g.illinois.edu/ { Homework solutions • http://www.math.illinois.edu/~murphyrf/teaching/M220/ { Tests and quizzes in my previous courses • Section 3.10 (Linear Approximation and Differentials) { Be able to use a tangent line (or differentials) in order to approximate the value of a function near the point of tangency. • Section 4.2 (The Mean Value Theorem) { Be able to precisely state The Mean Value Theorem and Rolle's Theorem. { Be able to decide when functions satisfy the conditions of these theorems. If a function does satisfy the conditions, then be able to find the value of c guaranteed by the theorems. { Be able to use The Mean Value Theorem, Rolle's Theorem, or earlier important the- orems such as The Intermediate Value Theorem to prove some other fact. In the homework these often involved roots, solutions, x-intercepts, or intersection points. • Section 4.8 (Newton's Method) { Understand the graphical basis for Newton's Method (that is, use the point where the tangent line crosses the x-axis as your next estimate for a root of a function). { Be able to apply Newton's Method to approximate roots, solutions, x-intercepts, or intersection points. -
Curl, Divergence and Laplacian
Curl, Divergence and Laplacian What to know: 1. The definition of curl and it two properties, that is, theorem 1, and be able to predict qualitatively how the curl of a vector field behaves from a picture. 2. The definition of divergence and it two properties, that is, if div F~ 6= 0 then F~ can't be written as the curl of another field, and be able to tell a vector field of clearly nonzero,positive or negative divergence from the picture. 3. Know the definition of the Laplace operator 4. Know what kind of objects those operator take as input and what they give as output. The curl operator Let's look at two plots of vector fields: Figure 1: The vector field Figure 2: The vector field h−y; x; 0i: h1; 1; 0i We can observe that the second one looks like it is rotating around the z axis. We'd like to be able to predict this kind of behavior without having to look at a picture. We also promised to find a criterion that checks whether a vector field is conservative in R3. Both of those goals are accomplished using a tool called the curl operator, even though neither of those two properties is exactly obvious from the definition we'll give. Definition 1. Let F~ = hP; Q; Ri be a vector field in R3, where P , Q and R are continuously differentiable. We define the curl operator: @R @Q @P @R @Q @P curl F~ = − ~i + − ~j + − ~k: (1) @y @z @z @x @x @y Remarks: 1. -
Calculus Terminology
AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential -
Calculus Formulas and Theorems
Formulas and Theorems for Reference I. Tbigonometric Formulas l. sin2d+c,cis2d:1 sec2d l*cot20:<:sc:20 +.I sin(-d) : -sitt0 t,rs(-//) = t r1sl/ : -tallH 7. sin(A* B) :sitrAcosB*silBcosA 8. : siri A cos B - siu B <:os,;l 9. cos(A+ B) - cos,4cos B - siuA siriB 10. cos(A- B) : cosA cosB + silrA sirrB 11. 2 sirrd t:osd 12. <'os20- coS2(i - siu20 : 2<'os2o - I - 1 - 2sin20 I 13. tan d : <.rft0 (:ost/ I 14. <:ol0 : sirrd tattH 1 15. (:OS I/ 1 16. cscd - ri" 6i /F tl r(. cos[I ^ -el : sitt d \l 18. -01 : COSA 215 216 Formulas and Theorems II. Differentiation Formulas !(r") - trr:"-1 Q,:I' ]tra-fg'+gf' gJ'-,f g' - * (i) ,l' ,I - (tt(.r))9'(.,') ,i;.[tyt.rt) l'' d, \ (sttt rrJ .* ('oqI' .7, tJ, \ . ./ stll lr dr. l('os J { 1a,,,t,:r) - .,' o.t "11'2 1(<,ot.r') - (,.(,2.r' Q:T rl , (sc'c:.r'J: sPl'.r tall 11 ,7, d, - (<:s<t.r,; - (ls(].]'(rot;.r fr("'),t -.'' ,1 - fr(u") o,'ltrc ,l ,, 1 ' tlll ri - (l.t' .f d,^ --: I -iAl'CSllLl'l t!.r' J1 - rz 1(Arcsi' r) : oT Il12 Formulas and Theorems 2I7 III. Integration Formulas 1. ,f "or:artC 2. [\0,-trrlrl *(' .t "r 3. [,' ,t.,: r^x| (' ,I 4. In' a,,: lL , ,' .l 111Q 5. In., a.r: .rhr.r' .r r (' ,l f 6. sirr.r d.r' - ( os.r'-t C ./ 7. /.,,.r' dr : sitr.i'| (' .t 8. tl:r:hr sec,rl+ C or ln Jccrsrl+ C ,f'r^rr f 9. -
10A– Three Generalizations of the Fundamental Theorem of Calculus MATH 22C
10A– Three Generalizations of the Fundamental Theorem of Calculus MATH 22C 1. Introduction In the next four sections we present applications of the three generalizations of the Fundamental Theorem of Cal- culus (FTC) to three space dimensions (x, y, z) 3,a version associated with each of the three linear operators,2R the Gradient, the Curl and the Divergence. Since much of classical physics is framed in terms of these three gener- alizations of FTC, these operators are often referred to as the three linear first order operators of classical physics. The FTC in one dimension states that the integral of a function over a closed interval [a, b] is equal to its anti- derivative evaluated between the endpoints of the interval: b f 0(x)dx = f(b) f(a). − Za This generalizes to the following three versions of the FTC in two and three dimensions. The first states that the line integral of a gradient vector field F = f along a curve , (in physics the work done by F) is exactlyr equal to the changeC in its potential potential f across the endpoints A, B of : C F Tds= f(B) f(A). (1) · − ZC The second, called Stokes Theorem, says that the flux of the Curl of a vector field F through a two dimensional surface in 3 is the line integral of F around the curve that S R 1 C 2 bounds : S CurlF n dσ = F Tds (2) · · ZZS ZC And the third, called the Divergence Theorem, states that the integral of the Divergence of F over an enclosed vol- ume is equal to the flux of F outward through the two dimensionalV closed surface that bounds : S V DivF dV = F n dσ. -
The Mean Value Theorem
Math 1A: introduction to functions and calculus Oliver Knill, 2014 Lecture 16: The mean value theorem In this lecture, we look at the mean value theorem and a special case called Rolle's theorem. Unlike the intermediate value theorem which applied for continuous functions, the mean value theorem involves derivatives. We assume therefore today that all functions are differentiable unless specified. Mean value theorem: Any interval (a; b) contains a point x such that f(b) − f(a) f 0(x) = : b − a f b -f a H L H L b-a Here are a few examples which illustrate the theorem: 1 Verify with the mean value theorem that the function f(x) = x2 +4 sin(πx)+5 has a point where the derivative is 1. Solution. Since f(0) = 5 and f(1) = 6 we see that (f(1) − f(0))=(1 − 0) = 5. 2 Verify that the function f(x) = 4 arctan(x)/π − cos(πx) has a point where the derivative is 3. Solution. We have f(0) = −1 and f(1) = 2. Apply the mean value theorem. 3 A biker drives with velocity f 0(t) at position f(b) at time b and at position a at time a. The value f(b) − f(a) is the distance traveled. The fraction [f(b) − f(a)]=(b − a) is the average speed. The theorem tells that there was a time when the bike had exactly the average speed. p 2 4 The function f(x) = 1 − x has a graph on (−1; 1) on which every possible slope isp taken. -
External Aerodynamics of the Magnetosphere
’ .., NASA TECHNICAL NOTE NASA TN- I- &: I -I LOAN COPY: R AFWL (W KIRTLAND AFI EXTERNAL AERODYNAMICS OF THE MAGNETOSPHERE by John R. Spreiter, A Zbertu Y. A Zksne, and Audrey L. Summers Awes Research Center Moffett Fie@ CuZ$ NATIONAL AERONAUTICS AND SPACE ADMINISTRATION WASHINGTON, D. C. JUNE 1968 i TECH LIBRARY KAFB, NM Illll1lllll111lll lllll lilll llll lllll 1111 Ill 0133359 EXTERNAL AERODYNAMICS OF THE MAGNETOSPHERE By John R. Spreiter, Alberta Y. Alksne, and Audrey L. Summers Ames Research Center Moffett Field, Calif. NATIONAL AERONAUTICS AND SPACE ADMINISTRATION For sale by the Clearinghouse for Federal Scientific and Technical Information Springfield, Virginia 22151 - CFSTl price $3.00 EXTERNAL AERODYNAMICS OF THE MAGNETOSPHERE* By John R. Spreiter, Alberta Y. Alksne, and Audrey L. Summers Ames Research Center SUMMARY A comprehensive survey is given of the continuum fluid theory of the solar wind and its interaction with the Earth's magnetic field, and the rela- tion between the calculated results and those actually measured in space. A unified basis for the entire discussion is provided by the equations of mag- netohydrodynamics, augmented by relations from kinetic theory for certain small-scale details of the flow. While the full complexity of magnetohydrodynamics is required for the formulation of the model and the establishment of the proper conditions to apply at the magnetosphere boundary, it is shown that the magnetic field actually experienced in space is usually sufficiently small that an adequate approximation to the solution can be obtained by first solving the simpler equations of gasdynamics for the flow and then using the results to calculate the deformation of the magnetic field.