DOCSLIB.ORG

Fundamental Theorems

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Fundamental Theorems Chapter 5 Fundamental Theorems 5.1 Revisiting the 1D Fundamental Theorem of Calculus Key Ideas. Alternate take on the derivative of a function f : R R. • ! – The function f determines a one-dimensional vector field V~ = f = f@ = fi. h i x – Let Br =[p r, p + r] be the interval of “radius” r centered at point p . ⇤ − ⇤ ⇤ – The boundary of Br consists of two points p + r and p r.The“outwardunitnormal” ⇤ ⇤ − to Br is given by @x at x = p + r Nˆ = ⇤ ( @x at x = p r. − ⇤ − – The outflux of V~ across the boundary of Br is V~ Nˆ = f(p + r) f(p r). · ⇤ − ⇤ − boundaryX points f(p + r) f(p r) – The outflux per unit size is ⇤ − ⇤ − 2r f(p + r) f(p r) – Linear approximation gives us ⇤ − ⇤ − = f 0(p )+error, where error 0. 2r ⇤ ! Thus the derivative of f at p is the limit of the “e↵ect per unit size” of the function. • ⇤ The Fundamental Theorem of Calculus can be obtained by summing over small intervals. • Our plan is now to find notions of derivatives, and corresponding fundamental theorems, for • various quantities. 73 74 CHAPTER 5. FUNDAMENTAL THEOREMS 5.2 Fundamental Theorem for Gradients Key Ideas. Consider a function f : Rn R and a path P (t)=(x(t),y(t),...) defined for a t b. Let • ! A = P (a) and B = P (b). We have f P :[a, b] R. By the 1D Fundamental Theorem of Calculus we have ◦ ! b (f P )0(t) dt = f(A) f(B). ◦ − Za From the Chain Rule we have • @f dx @x(P (t)) dt @f dy (f P )0(t)=[Df(P (t))]P 0(t)=0 @y (P (t))1 0 dt 1 ◦ · . B . C B . C B . C B . C @ A @ A Recall that the gradient of a function f: • @f @f grad(f)= , ,... @x @y ⌧ Geometric idea of gradient: The gradient of a function measures the total change of a function. • If Aˆ is any unit vector, then grad(f) Aˆ tells us the extent to which f changes in that direction. (This is the same as the directional derivative.)· The gradient vector points in the direction of largest increase, with magnitude equal to the rate of change in that direction. Di↵erentiation rules: • – Scaling rule: grad(cf)=c grad(f) if c is a constant – Addition rule: grad(f + g) = grad(f) + grad(g) – Product rule: grad fg = grad(f)g + f grad(g) Fundamental Theorem of Calculus for gradients:Supposef : Rn R is smooth1 and • that C is a parametrized path from point A to point B. Then ! grad(f) Tdsˆ = f(B) f(A). · − ZC Exercises. 1. Compute the gradient for the following functions: (a) f(x, y)=x2 + y2 (b) f(x, y)=x2 y2 − (c) f(x, y)=ln(x2 + y2) 1In this course smooth means that the object is continuous and that derivatrives of the object are also continuous. 5.2. FUNDAMENTAL THEOREM FOR GRADIENTS 75 (d) f(x, y, z)= x2 + y2 + z2 2. Can you find a functionp f so that. (a) . grad(f)= 2, 3 ? h i (b) . grad(f)= x, y ? h i (c) . grad(f)= y, x ? h i (d) . grad(f)= y, x ? h − i 3. Let C be the path traveling counter-clockwise along the unit circle from (1, 0) to ( 1, 0). Let L be the path traveling along the x axis from (1, 0) to ( 1, 0). − − (a) Find parametrizations for C and L. (b) Compute x, y Tdsˆ by direct methods. C h i· (c) Compute R x, y Tdsˆ by direct methods. Lh i· (d) Find a function f such that grad(f)= x, y . R h i (e) Compute x, y Tdsˆ using the Fundamental Theorem for Gradients. C h i· (f) Compute R x, y Tdsˆ using the Fundamental Theorem for Gradients. Lh i· (g) Do your computationsR agree? Explain why/why not. 4. Let C be the path traveling counter-clockwise along the unit circle from (1, 0) to ( 1, 0). Let − C be the path traveling clockwise along the unit circle from (1, 0) to ( 1, 0). − e(a) Find parametrizations of C and C. (b) Compute y, x Tdsˆ by direct methods. C h − i· e (c) Compute R y, x Tdsˆ by direct methods. C h − i· (d) Do your computations agree? Explain why/why not. R e 76 CHAPTER 5. FUNDAMENTAL THEOREMS 5.3 Divergence of a vector field Key Ideas. Geometric idea of divergence: The divergence of vector field V~ at the point P measures the • expansion of V~ at P ⇤ ⇤ Definition of divergence using limits: Let Br be the solid ball of radius r centered at point P • ⇤ and Sr be the boundary of Br. 1 1 div V~ (P )=lim V~ NdAˆ and/or div V~ (P )=lim V~ Ndsˆ ⇤ r 0 B · ⇤ r 0 B · ! | r| ZZSr ! | r| ZSr Computation in Cartesian coordinates: Suppose V~ = V ,V ,... Then • h x y i @V @V div V~ = x + y + .... @x @y Di↵erentiation rules: • – Scaling rule: div(cV~ )=c div V~ if c is a constant – Addition rule: div(V~ + W~ )=divV~ +divW~ – Product rule: div(fV~ ) = grad(f) V~ + f div V~ if f is a function · Exercises. 1. For each vector field do the following: make a sketch of the vector field by hand, describe in words what the vector field does, compute the divergence of the vector field, comment on how your computation fits with your earlier description. (a) V~ = x, y h i (b) V~ = y, x h i (c) V~ = 3, 2 h − i (d) V~ = x , y h x2+y2 x2+y2 i (e) V~ = x y, x + y h − i (f) V~ = x + y, x y h − i (g) V~ = y2,y2 h i 2. Suppose f(x, y)=x2 y2 and V~ = x + y, x +2y . − h i (a) Compute fV~ and then compute div(fV~ ). (b) Use the product rule to compute div(fV~ ). (c) Prove the product rule formula for divergence. 5.3. DIVERGENCE OF A VECTOR FIELD 77 3. The Laplacian of a function f is defined by div(grad f). Compute the Laplacian of the following functions. (a) f(x, y)=2x 3y − (b) f(x, y)=x2 + y2 (c) f(x, y)=x2 y2 − (d) f(x, y) = cos(x)sin(y) 1 2 2 (e) f(x, y)= 2 ln(x + y ) 1 (f) f(x, y, z)= x2 + y2 + z2 What property ofp the function does the Laplacian seem to measure? 4. Consider the vector field given in spherical coordinates by 1 x y z V~ = @ = , , r2 r (x2 + y2 + z2)3/2 (x2 + y2 + z2)3/2 (x2 + y2 + z2)3/2 D E and concentric spheres Sr of radius r centered at the origin. (a) Compute the outward flux V~ NdAˆ using the definition of the flux integral; Sr · (b) Find the flux rate RR V~ NdAˆ lim Sr · r 0 Volume(B ) ! RR r where Br denotes the ball enclosed by Sr. (c) Compute the divergence of the vector field V~ . (d) What is the outward flux rate of V~ at the origin? What about anywhere other than the origin? 5. (Optional Challenge Problem) In this problem you consider the possibility of defining diver- gence of a vector field V~ (x, y) using a square, rather than a ball. Fill in the boxes in the argument below. (a) For simplicity we work in two dimensions. Write V~ (x, y)= Vx(x, y),Vy(x, y) .Fixa point (x ,y ). For small x and y we have h i ⇤ ⇤ @V @V V~ (x, y) V~ (x ,y )+(x x ) x (x ,y ), y (x ,y ) ⇡ ⇤ ⇤ − ⇤ @x ⇤ ⇤ @x ⇤ ⇤ ⌧ +(y y ) , − ⇤ * + 2 (b) Let Br denote the interior of the square in R having corners at (x + r, y + r), (x ⇤ ⇤ ⇤ − r, y + r), (x r, y r), (x + r, y r). Let Sr be the boundary of Br, oriented counter-clockwise.⇤ ⇤ − ⇤ − ⇤ ⇤ − 78 CHAPTER 5. FUNDAMENTAL THEOREMS Draw a picture showing Br and Sr. The area of Br is (c) The outward unit normal Nˆ along Sr is given by 1, 0 when x = x + r, h i ⇤ 8 when y = y + r, > ⇤ > ˆ > N = > > when , <> > > when , > > > (d) The integral V~ Ndsˆ can:> be approximated by Sr · r R r − V~ (x + r, y + t) 1, 0 dt + 0, 1 dt t= r ⇤ ⇤ ·h i t=r ·h i Z − Z r − + dt + . Zt=r (e) Using the approximation from the first part of this problem we we find that r @V @V V~ Ndsˆ V~ (x ,y )+r x (x ,y )+t x (x ,y ) dt S · ⇡ t= r ⇤ ⇤ @x ⇤ ⇤ @y ⇤ ⇤ Z r Z − ✓ ◆ r − + V~ (x ,y )+t + r dt 0 ⇤ ⇤ 1 Zt=r B C @ A r − V~ (x ,y )+r t dt − 0 ⇤ ⇤ − 1 Zt=r B C @ A r dt − t= r Z − (f) The previous integral simplifies to @V V~ Ndsˆ x (x ,y )+ . · ⇡ 0 @x ⇤ ⇤ 1 ZSr B C @ A 5.3. DIVERGENCE OF A VECTOR FIELD 79 Thus V~ Ndsˆ @V lim Sr · = x (x ,y )+ r 0 @x ⇤ ⇤ ! R and we are done! 80 CHAPTER 5. FUNDAMENTAL THEOREMS 5.4 Divergence Theorems Key Ideas. Two-dimensional Divergence Theorem. Let D be a bounded domain in R2, let C be the • boundary of D, and let Nˆ be the outward-pointing unit normal along C. Then for any smooth vector field V~ defined in D we have div VdA~ = V~ Nds.ˆ · ZZD ZC The two-dimensional divergence theorem is equivalent to Green’s Theorem.
Load more

Recommended publications
  • Lecture 9: Partial Derivatives
    Math S21a: Multivariable calculus Oliver Knill, Summer 2016 Lecture 9: Partial derivatives ∂ If f(x,y) is a function of two variables, then ∂x f(x,y) is defined as the derivative of the function g(x) = f(x,y) with respect to x, where y is considered a constant. It is called the partial derivative of f with respect to x. The partial derivative with respect to y is defined in the same way. ∂ We use the short hand notation fx(x,y) = ∂x f(x,y). For iterated derivatives, the notation is ∂ ∂ similar: for example fxy = ∂x ∂y f. The meaning of fx(x0,y0) is the slope of the graph sliced at (x0,y0) in the x direction. The second derivative fxx is a measure of concavity in that direction. The meaning of fxy is the rate of change of the slope if you change the slicing. The notation for partial derivatives ∂xf,∂yf was introduced by Carl Gustav Jacobi. Before, Josef Lagrange had used the term ”partial differences”. Partial derivatives fx and fy measure the rate of change of the function in the x or y directions. For functions of more variables, the partial derivatives are defined in a similar way. 4 2 2 4 3 2 2 2 1 For f(x,y)= x 6x y + y , we have fx(x,y)=4x 12xy ,fxx = 12x 12y ,fy(x,y)= − − − 12x2y +4y3,f = 12x2 +12y2 and see that f + f = 0. A function which satisfies this − yy − xx yy equation is also called harmonic. The equation fxx + fyy = 0 is an example of a partial differential equation: it is an equation for an unknown function f(x,y) which involves partial derivatives with respect to more than one variables.
    [Show full text]
  • Differentiation Rules (Differential Calculus)
    Differentiation Rules (Differential Calculus) 1. Notation The derivative of a function f with respect to one independent variable (usually x or t) is a function that will be denoted by D f . Note that f (x) and (D f )(x) are the values of these functions at x. 2. Alternate Notations for (D f )(x) d d f (x) d f 0 (1) For functions f in one variable, x, alternate notations are: Dx f (x), dx f (x), dx , dx (x), f (x), f (x). The “(x)” part might be dropped although technically this changes the meaning: f is the name of a function, dy 0 whereas f (x) is the value of it at x. If y = f (x), then Dxy, dx , y , etc. can be used. If the variable t represents time then Dt f can be written f˙. The differential, “d f ”, and the change in f ,“D f ”, are related to the derivative but have special meanings and are never used to indicate ordinary differentiation. dy 0 Historical note: Newton used y,˙ while Leibniz used dx . About a century later Lagrange introduced y and Arbogast introduced the operator notation D. 3. Domains The domain of D f is always a subset of the domain of f . The conventional domain of f , if f (x) is given by an algebraic expression, is all values of x for which the expression is defined and results in a real number. If f has the conventional domain, then D f usually, but not always, has conventional domain. Exceptions are noted below.
    [Show full text]
  • 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
    [Show full text]
  • CHAPTER 3: Derivatives
    CHAPTER 3: Derivatives 3.1: Derivatives, Tangent Lines, and Rates of Change 3.2: Derivative Functions and Differentiability 3.3: Techniques of Differentiation 3.4: Derivatives of Trigonometric Functions 3.5: Differentials and Linearization of Functions 3.6: Chain Rule 3.7: Implicit Differentiation 3.8: Related Rates • Derivatives represent slopes of tangent lines and rates of change (such as velocity). • In this chapter, we will define derivatives and derivative functions using limits. • We will develop short cut techniques for finding derivatives. • Tangent lines correspond to local linear approximations of functions. • Implicit differentiation is a technique used in applied related rates problems. (Section 3.1: Derivatives, Tangent Lines, and Rates of Change) 3.1.1 SECTION 3.1: DERIVATIVES, TANGENT LINES, AND RATES OF CHANGE LEARNING OBJECTIVES • Relate difference quotients to slopes of secant lines and average rates of change. • Know, understand, and apply the Limit Definition of the Derivative at a Point. • Relate derivatives to slopes of tangent lines and instantaneous rates of change. • Relate opposite reciprocals of derivatives to slopes of normal lines. PART A: SECANT LINES • For now, assume that f is a polynomial function of x. (We will relax this assumption in Part B.) Assume that a is a constant. • Temporarily fix an arbitrary real value of x. (By “arbitrary,” we mean that any real value will do). Later, instead of thinking of x as a fixed (or single) value, we will think of it as a “moving” or “varying” variable that can take on different values. The secant line to the graph of f on the interval []a, x , where a < x , is the line that passes through the points a, fa and x, fx.
    [Show full text]
  • Antiderivatives 307
    4100 AWL/Thomas_ch04p244-324 8/20/04 9:02 AM Page 307 4.8 Antiderivatives 307 4.8 Antiderivatives We have studied how to find the derivative of a function. However, many problems re- quire that we recover a function from its known derivative (from its known rate of change). For instance, we may know the velocity function of an object falling from an initial height and need to know its height at any time over some period. More generally, we want to find a function F from its derivative ƒ. If such a function F exists, it is called an anti- derivative of ƒ. Finding Antiderivatives DEFINITION Antiderivative A function F is an antiderivative of ƒ on an interval I if F¿sxd = ƒsxd for all x in I. The process of recovering a function F(x) from its derivative ƒ(x) is called antidiffer- entiation. We use capital letters such as F to represent an antiderivative of a function ƒ, G to represent an antiderivative of g, and so forth. EXAMPLE 1 Finding Antiderivatives Find an antiderivative for each of the following functions. (a) ƒsxd = 2x (b) gsxd = cos x (c) hsxd = 2x + cos x Solution (a) Fsxd = x2 (b) Gsxd = sin x (c) Hsxd = x2 + sin x Each answer can be checked by differentiating. The derivative of Fsxd = x2 is 2x. The derivative of Gsxd = sin x is cos x and the derivative of Hsxd = x2 + sin x is 2x + cos x. The function Fsxd = x2 is not the only function whose derivative is 2x. The function x2 + 1 has the same derivative.
    [Show full text]
  • Numerical Differentiation
    Jim Lambers MAT 460/560 Fall Semester 2009-10 Lecture 23 Notes These notes correspond to Section 4.1 in the text. Numerical Differentiation We now discuss the other fundamental problem from calculus that frequently arises in scientific applications, the problem of computing the derivative of a given function f(x). Finite Difference Approximations 0 Recall that the derivative of f(x) at a point x0, denoted f (x0), is defined by 0 f(x0 + h) − f(x0) f (x0) = lim : h!0 h 0 This definition suggests a method for approximating f (x0). If we choose h to be a small positive constant, then f(x + h) − f(x ) f 0(x ) ≈ 0 0 : 0 h This approximation is called the forward difference formula. 00 To estimate the accuracy of this approximation, we note that if f (x) exists on [x0; x0 + h], 0 00 2 then, by Taylor's Theorem, f(x0 +h) = f(x0)+f (x0)h+f ()h =2; where 2 [x0; x0 +h]: Solving 0 for f (x0), we obtain f(x + h) − f(x ) f 00() f 0(x ) = 0 0 + h; 0 h 2 so the error in the forward difference formula is O(h). We say that this formula is first-order accurate. 0 The forward-difference formula is called a finite difference approximation to f (x0), because it approximates f 0(x) using values of f(x) at points that have a small, but finite, distance between them, as opposed to the definition of the derivative, that takes a limit and therefore computes the derivative using an “infinitely small" value of h.
    [Show full text]
  • Review of Differentiation and Integration
    Schreyer Fall 2018 Review of Differentiation and Integration for Ordinary Differential Equations In this course you will be expected to be able to differentiate and integrate quickly and accurately. Many students take this course after having taken their previous course many years ago, at another institution where certain topics may have been omitted, or just feel uncomfortable with particular techniques. Because understanding this material is so important to being successful in this course, we have put together this brief review packet. In this packet you will find sample questions and a brief discussion of each topic. If you find the material in this pamphlet is not sufficient for you, it may be necessary for you to use additional resources, such as a calculus textbook or online materials. Because this is considered prerequisite material, it is ultimately your responsibility to learn it. The topics to be covered include Differentiation and Integration. 1 Differentiation Exercises: 1. Find the derivative of y = x3 sin(x). ln(x) 2. Find the derivative of y = cos(x) . 3. Find the derivative of y = ln(sin(e2x)). Discussion: It is expected that you know, without looking at a table, the following differentiation rules: d [(kx)n] = kn(kx)n−1 (1) dx d h i ekx = kekx (2) dx d 1 [ln(kx)] = (3) dx x d [sin(kx)] = k cos kx (4) dx d [cos(kx)] = −k sin x (5) dx d [uv] = u0v + uv0 (6) dx 1 d u u0v − uv0 = (7) dx v v2 d [u(v(x))] = u0(v)v0(x): (8) dx We put in the constant k into (1) - (5) because a very common mistake to make is something d e2x like: e2x = (when the correct answer is 2e2x).
    [Show full text]
  • Just the Definitions, Theorem Statements, Proofs On
    MATH 3333{INTERMEDIATE ANALYSIS{BLECHER NOTES 55 Abbreviated notes version for Test 3: just the definitions, theorem statements, proofs on the list. I may possibly have made a mistake, so check it. Also, this is not intended as a REPLACEMENT for your classnotes; the classnotes have lots of other things that you may need for your understanding, like worked examples. 6. The derivative 6.1. Differentiation rules. Definition: Let f :(a; b) ! R and c 2 (a; b). If the limit lim f(x)−f(c) exists and is finite, then we say that f is differentiable at x!c x−c 0 df c, and we write this limit as f (c) or dx (c). This is the derivative of f at c, and also obviously equals lim f(c+h)−f(c) by setting x = c + h or h = x − c. If f is h!0 h differentiable at every point in (a; b), then we say that f is differentiable on (a; b). Theorem 6.1. If f :(a; b) ! R is differentiable at at a point c 2 (a; b), then f is continuous at c. Proof. If f is differentiable at c then f(x) − f(c) f(x) = (x − c) + f(c) ! f 0(c)0 + f(c) = f(c); x − c as x ! c. So f is continuous at c. Theorem 6.2. (Calculus I differentiation laws) If f; g :(a; b) ! R is differentiable at a point c 2 (a; b), then (1) f(x) + g(x) is differentiable at c and (f + g)0(c) = f 0(c) + g0(c).
    [Show full text]
  • MATH 162: Calculus II Differentiation
    MATH 162: Calculus II Framework for Mon., Jan. 29 Review of Differentiation and Integration Differentiation Definition of derivative f 0(x): f(x + h) − f(x) f(y) − f(x) lim or lim . h→0 h y→x y − x Differentiation rules: 1. Sum/Difference rule: If f, g are differentiable at x0, then 0 0 0 (f ± g) (x0) = f (x0) ± g (x0). 2. Product rule: If f, g are differentiable at x0, then 0 0 0 (fg) (x0) = f (x0)g(x0) + f(x0)g (x0). 3. Quotient rule: If f, g are differentiable at x0, and g(x0) 6= 0, then 0 0 0 f f (x0)g(x0) − f(x0)g (x0) (x0) = 2 . g [g(x0)] 4. Chain rule: If g is differentiable at x0, and f is differentiable at g(x0), then 0 0 0 (f ◦ g) (x0) = f (g(x0))g (x0). This rule may also be expressed as dy dy du = . dx x=x0 du u=u(x0) dx x=x0 Implicit differentiation is a consequence of the chain rule. For instance, if y is really dependent upon x (i.e., y = y(x)), and if u = y3, then d du du dy d (y3) = = = (y3)y0(x) = 3y2y0. dx dx dy dx dy Practice: Find d x d √ d , (x2 y), and [y cos(xy)]. dx y dx dx MATH 162—Framework for Mon., Jan. 29 Review of Differentiation and Integration Integration The definite integral • the area problem • Riemann sums • definition Fundamental Theorem of Calculus: R x I: Suppose f is continuous on [a, b].
    [Show full text]
  • ATMS 310 Math Review Differential Calculus the First Derivative of Y = F(X)
    ATMS 310 Math Review Differential Calculus dy The first derivative of y = f(x) = = f `(x) dx d2 x The second derivative of y = f(x) = = f ``(x) dy2 Some differentiation rules: dyn = nyn-1 dx d() uv dv du = u( ) + v( ) dx dx dx ⎛ u ⎞ ⎛ du dv ⎞ d⎜ ⎟ ⎜v − u ⎟ v dx dx ⎝ ⎠ = ⎝ ⎠ dx v2 dy ⎛ dy dz ⎞ Chain Rule If y = f(z) and z = f(x), = ⎜ *⎟ dx ⎝ dz dx ⎠ Total vs. Partial Derivatives The rate of change of something following a fluid element (e.g. change in temperature in the atmosphere) is called the Langrangian rate of change: d / dt or D / Dt The rate of change of something at a fixed point is called the Eulerian (oy – layer’ – ian) rate of change: ∂/∂t, ∂/∂x, ∂/∂y, ∂/∂z The total rate of change of a variable is the sum of the local rate of change plus the 3- dimensional advection of that variable: d() ∂( ) ∂( ) ∂( ) ∂( ) = + u + v + w dt∂ t ∂x ∂y ∂z (1) (2) (3) (4) Where (1) is the Eulerian rate of change, and terms (2) – (4) is the 3-dimensional advection of the variable (u = zonal wind, v = meridional wind, w = vertical wind) Vectors A scalar only has a magnitude (e.g. temperature). A vector has magnitude and direction (e.g. wind). Vectors are usually represented in bold font. The wind vector is specified as V or r sometimes as V i, j, and k known as unit vectors. They have a magnitude of 1 and point in the x (i), y (j), and z (k) directions.
    [Show full text]
  • Efficient Computation of Sparse Higher Derivative Tensors⋆
    Efficient Computation of Sparse Higher Derivative Tensors? Jens Deussen and Uwe Naumann RWTH Aachen University, Software and Tools for Computational Engineering, Aachen, Germany fdeussen,[email protected] Abstract. The computation of higher derivatives tensors is expensive even for adjoint algorithmic differentiation methods. In this work we in- troduce methods to exploit the symmetry and the sparsity structure of higher derivatives to considerably improve the efficiency of their compu- tation. The proposed methods apply coloring algorithms to two-dimen- sional compressed slices of the derivative tensors. The presented work is a step towards feasibility of higher-order methods which might benefit numerical simulations in numerous applications of computational science and engineering. Keywords: Adjoints · Algorithmic Differentiation · Coloring Algorithm · Higher Derivative Tensors · Recursive Coloring · Sparsity. 1 Introduction The preferred numerical method to compute derivatives of a computer program is Algorithmic Differentiation (AD) [1,2]. This technique produces exact derivatives with machine accuracy up to an arbitrary order by exploiting elemental symbolic differentiation rules and the chain rule. AD distinguishes between two basic modes: the forward mode and the reverse mode. The forward mode builds a directional derivative (also: tangent) code for the computation of Jacobians at a cost proportional to the number of input parameters. Similarly, the reverse mode builds an adjoint version of the program but it can compute the Jacobian at a cost that is proportional to the number of outputs. For that reason, the adjoint method is advantageous for problems with a small set of output parameters, as it occurs frequently in many fields of computational science and engineering.
    [Show full text]
  • AP Calculus BC Scope & Sequence
    AP Calculus BC Scope & Sequence Grading Period Unit Title Learning Targets Throughout the *Apply mathematics to problems in everyday life School Year *Use a problem-solving model that incorporates analyzing information, formulating a plan, determining a solution, justifying the solution and evaluating the reasonableness of the solution *Select tools to solve problems *Communicate mathematical ideas, reasoning and their implications using multiple representations *Create and use representations to organize, record and communicate mathematical ideas *Analyze mathematical relationships to connect and communicate mathematical ideas *Display, explain and justify mathematical ideas and arguments First Grading Preparation for Pre Calculus Review Period Calculus Limits and Their ● An introduction to limits, including an intuitive understanding of the limit process and the Properties formal definition for limits of functions ● Using graphs and tables of data to determine limits of functions and sequences ● Properties of limits ● Algebraic techniques for evaluating limits of functions ● Comparing relative magnitudes of functions and their rates of change (comparing exponential growth, polynomial growth and logarithmic growth) ● Continuity and one-sided limits ● Geometric understanding of the graphs of continuous functions ● Intermediate Value Theorem ● Infinite limits ● Understanding asymptotes in terms of graphical behavior ● Using limits to find the asymptotes of a function, vertical and horizontal Differentiation ● Tangent line to a curve and
    [Show full text]