Math 275, section 002 (Ultman) Spring 2012 FINAL EXAM REVIEW The final exam will be held on Wednesday 9 May from 8:00 10:00am in our regular classroom. You will be allowed both sides of two 8.5 11 sheets of paper for handwritten notes and a scientific calculator. In addition, you may use the sheet of basic derivatives and integrals posted on the course website. No other notes or technology will be allowed for this exam. This exam will be over all material covered in class this semester. A significant percentage (somewhere between 50 80%) will address current material, corresponding to chapter 15 in the text. The remainder will be review material from earlier in the semster. This review includes suggested review problems. For problems that will help you work on basic skills, go to the Supplemental exercises link on the course website. For additional review problems, look at the previous exam reviews, past homework assignments, worksheets, exams, examples from the course notes, etc. CURRENT MATERIAL: Vector Calculus. Suggested review problems: sec. 15.1 # 27, 29, 34 sec. 15.2 # 19, 25, 35 sec. 15.3 # 3, 29, 32, 35, 36 sec. 15.5 # 17, 19, 21, 23, 25 sec. 15.6 # 6, 15, 23, 24, 28, 37 sec. 15.7, 15.8: problems in Assignment 13.
VECTOR FIELDS & LINE INTEGRALS Key Words: line integral; scalar line element ds; infinitesimal displacement vector d r; work; flow/circulation; vector fields; conservative vector fields; Fundamental Theorem of Line Integrals. Key Ideas: A vector field is a function that assigns a vector to each point in its domain. A vector field is continuous if its component functions are continuous. Important examples of vector fields include: the gradient of a function; the rectangular basis vectors î, ĵ and ˆk (these are constant fields); the planar radial field x î+y ĵ = r cos θ î+r sin θ ĵ, and the unit radial field ˆr = (x î + y ĵ)/ x 2 + y 2 = cos θ î + sin θ ĵ; the radial field in 3-space x î + y ĵ + z ˆk, and the unit radial field ˆρ = (x î + y ĵ + z ˆk)/ x 2 + y 2 + z 2 ; the planar counter-clockwse spin field ˆθ = y î + x ĵ = r sin θ î + r cos θ ĵ and the unit spin field ˆθ = ( y î+x ĵ)/ x 2 + y 2 = sin θ î+ cos θ ĵ. Scalar line integrals generalize integrals of functions of a single variable to arbitrary (smooth) curves. The scalar line element ds (aka arc length element) can be thought of as an infinitesimal version of the Pythagorean theorem, ds 2 = dx 2 + dy 2 + dz 2. If C is a smooth curve and f is a real-valued function defined along the curve, the (scalar) line integral of f along C is f ds. Applications of scalar line integrals include: C length of curves (covered in sec 12.3); the mass, first moments and centers of mass, and second moments (moments of inertia) of one-dimensional objects (for example, thin wires). The infinitesimal vector d r is the vector differential of the position vector r, and represents an infinitesimal change in position that is, an infinitesimal displacement. The arc length element ds is the magnitude of the vector d r. When restricted to a smooth curve, d r is tangent to the curve, and hence is a scalar multiple of the unit tangent vector ˆT ; specifically, d r = ˆT ds. If r(t) is a smooth parameterization of a curve, then d r = ˆT ds = r (t) dt. 2
The component of a vector field F acting tangentially along an infinitesimal section of a curve is given by the dot product F ˆT ds = F d r. Vector line integrals add up these scaled projections along the curve: if F is continuous and C is smooth, the (vector) line integral of F along C is F d r. Depending on context, the line integral F d r has various C C interpretations. For example: If F is a force field, then F d r is the work performed by the field in moving an object through the displacement d r, and the line integral C F d r measures the work performed by the force in moving an object along the curve C. If F is a velocity field, the line integral measures the flow of the field along the curve (called circulation if the curve is a closed loop). The Fundamental Theorem of Line Integrals (FTLI) states that if F = f (that is, F is a gradient field) and C is a piecewise smooth curve, then the integral C F d r can be computed by evaluating the potential function f at the endpoints and subtracting the value at the initial point from the value at the terminal point (cf. the Fundamental Theorem of Calculus). A vector field F is conservative if line integrals C F d r are independent of path, that is: for any points A and B and any smooth curves C 1 and C 2 from A to B, C 1 F d r = C 2 F d r. The following statements are equivalent: F is conservative; line integrals are independent of path; line integrals around closed curves equal zero; F is (locally) a gradient field. If a vector field is continuously differentiable on a simply connected domain, then the curl test can be used to determine whether the field is conservative, since on a simply connected domain F = f if and only if curl F = 0. One method for finding a potential function f of a conservative field F (over a simply connected domain) is to integrate the component functions with respect to the appropriate variables and compare the results. For this exam, you are expected to be able to: * perform basic computations: given a smooth parameterization, compute d r and ds; set up and evaluate both scalar and vector line integrals; use the curl test to identify conservative vector fields; find potential functions for conservative vector fields 3
* know the statement of and be able to apply the FTLI; * recognize whether or not a vector-valued function r(t) parameterizes a given curve; * compute the mass, center of mass and first and second moments of a one-dimensional object; * compute work and flow (or circulation) along smooth curves; * recognize the equivalent definitions of conservative vector fields and be able to use them to analyze statements about line integrals; * understand how the geometric relationship between the field and the curve affects a line integral, and be able to apply this understanding to analyze statements about and diagrammatic representations of line integrals. SURFACE AND FLUX INTEGRALS Key Words: vector area element d A; scalar area element da; flux. Key Ideas: Surface integrals generalize double integrals over a region in the xy-plane to integrals defined on (possibly) non-planar surfaces. The scalar area element da is a measure of area on the surface. Given a smooth parameterization r(u, v), da is the area of the infinitesimal parallelogram spanned by the infinitesimal displacement vectors d r u and d r v, which can be computed by finding the magnitude of the cross product d r u d r v. Occasionally, da can be determined using purely geometric arguments. The vector area element da is a vector normal to a surface, with magnitude da = da. The component of a field F passing through (normal to) an infinitesimal region of a surface is given by the dot product F da = F ˆn da, where ˆn is a unit vector normal to the surface. Given a smooth parameterization r(u, v), the vector area element can be computed by da = ± d r u d r v. The flux of the field F through the oriented surface S is given by the integral F da. S For this exam, you are expected to be able to: * perform basic computations: given a smooth parameterization, compute d A and da; set up and evaluate both (scalar) surface integrals and flux integrals; 4
* recognize whether or not a vector-valued function r(u, v) parameterizes a given surface; * compute the mass, center of mass, and moments of inertia of a twodimensional object; * understand how the geometric relationship between the field and the surface affects a flux integral, and be able to apply this understanding to analyze statements about and diagrammatic representations of flux integrals. STOKES THEOREM AND THE DIVERGENCE THEOREM Key Words: curl; divergence; Stokes theorem; Green s Theorem; Divergence Theorem. Key Ideas: The curl of a vector field is a vector field that encodes the circulation density of the field. Circulation density in a given direction can be found by taking the dot product of the curl field with the unit vector in that direction (cf. the gradient). The dot product curl F d A is the circulation of the field F about an infinitesimal closed curve in the plane normal to d A. Stokes theorem states that, under appropriate conditions, the circulation of a field around a closed curve can be determined by finding the flux of the curl of the field through a surface bounded by the curve. If the curve and the surface both lie in a plane, the result is called Green s theorem. The divergence of a vector field at a point gives a measure of the flux density of a vector field. The (scalar) product div F dv is the flux of the field F through an infinitesimal closed surface. The Divergence Theorem states that, under appropriate conditions, the flux of a field through a closed oriented surface is equivalent to the integral of the divergence of the field over the solid region enclosed by the surface. It is always true that the curl of a gradient field is zero and the divergence of a curl field is zero (this can be checked by computation it s a good exercise). The converse is true when the field is once continuously differentiable over a simply connected domain: that is, over a simply connected domain, a field with continuous partial derivatives is a gradient field if its curl is zero and is a curl field if its divergence is zero. 5
For this exam, you are expected to be able to: * perform basic computations: compute the curl and divergence of a vector field; * know the picture motivating Stokes (and Green s) Theorem this picture connects curl, circulation around infinitesimal curves on the surface, and circulation around the boundary curve; * know the picture motivating the Divergence Theorem this picture connects divergence, flux through infinitesimal boxes on the interior, and flux through the bounding surface; * know the statements of Stokes, Green s and the Divergence Theorems; recognize the major similarities between these theorems, both with each other and also with the FTLI and the FTC; * apply Stokes, Green s and the Divergence Theorems as directed. summary of the three big theorems of vector calculus The results of the FTLI, Stokes theorem and the divergence theorem are very similar. Roughly speaking, they relate the interaction of a field with a bounding object to the interaction of a derivative of the field with the region being bounded (here, field includes scalar-valued functions, or scalar fields ). In fact, we could throw in the Fundamental theorem of calculus as well; the relationship between the FTC and the FTLI is analogous to that between Green s and Stokes theorems. Fundamental Theorem of Line Integrals : f(b) f(a) = (f) d r C Stokes Theorem : F d r = curlf da C S Divergence Theorem : F da = divf dv S V 6
REVIEW MATERIAL For a more detailed review of this material, see the reviews for the previous exams. Suggested review problems: ch. 11: sec. 11.1 # 64 sec. 11.2 # 42, 44 sec. 11.3 # 16, 23, 28, 30, 44 sec. 11.4 # 23, 28, 29, 31, 47 sec. 11.5 # 3, 10, 22, 23 ch. 12: sec. 12.1 # 8, 18, 23, 25 sec. 12.3 # 9, 15, 19 sec. 12.4 # 3, 9 sec. 12.5 # 1, 5, 17 19, 23 ch. 13: sec. 13.1 # 14 16, 31 36, 39, 40 sec. 13.3 # 39, 49, 56, 90 sec. 13.4 # 41, 42, 47, 48, 49 sec. 13.5 # 17, 21, 27, 30, 31, 33, 36 sec. 13.6 # 31, 50, 51, 54 sec. 13.7 # 41, 43, 44, 46, 47 ch. 14: sec. 14.2 # 53, 61, 77, 80 sec. 14.3 # 5, 15, 17, 20, 21 sec. 14.4 # 6, 21, 25, 33, 36, 38 sec. 14.5 # 21, 25, 36, 41 sec. 14.6 # 17, 25 sec. 14.7 # 11a&b, 31a, 39, 40, 41, 67. VECTORS Vectors are objects described by a scalar component (magnitude) and a direction. Two vectors can be added and a vector can be multiplied by a scalar. Vector addition and scalar multiplication can be used to find a parameterization of a line. The dot product of two vectors, which is a scalar, encodes information about the magnitudes of the vectors and the angle between them: v w = v w cos θ. The dot product is used to define orthogonality of vectors. 7
The dot product is also used to find vector projections. Applications of the dot product include: distance between points (this is equivalent to finding the magnitude of vectors); equations of spheres; equations planes in R 3 ; the work performed by a constant force over a straight-line displacement. The cross product is a vector. The magnitude of u v is the area of the parallelogram having sides u and v. The direction of u v is normal to the plane containing u and v, and is determined by the right-hand rule (if u and v are colinear or if either is the zero vector, then u v = 0). Geometrically, u v = u v sin θ, so the cross product can be used to detect when two vectors are parallel. Applications of the cross product include: area of triangles and parallelograms; finding a normal vector to a plane; computing torque. For this exam, you are expected to be able to: * perform basic computations: vector addition, scalar multiplication, cross and dot products, vector projections; * use vector operations to find paramerterizations of lines, normal vectors to planes, and equations of planes and spheres; * understand the geometric and algebraic properties of the dot and cross products. PARAMETERIZED CURVES A curve is parameterized if its coordinates have been written as functions of a single variable (the parameter). The parameterization is often expressed as a vector-valued function r(u). A parameterized curve is continuous (resp. differentiable, integrable) if its coordinate functions are continuous (resp. differentiable, integrable). A parameterization is smooth if it is continuously differentiable and the derivative is never the zero vector. The derivative of a parameterized curve is a vector tangent to the curve. If a parameterization represents the position of a particle in space as a function of time, the derivative is the velocity and the second derivative is the acceleration of the particle. The differential of a parameterized curve is the infinitesimal displacement vector along the curve. Applications of parameterized curves, their integrals and derivatives include: motion in space (velocity, acceleration); curvature; length of curves. 8
For this exam, you are expected to be able to: * perform basic computations: given a smooth parameterization, compute the velocity and acceleration, the unit tangent, normal and binormal vectors (the { ˆT, ˆN, ˆB} frame), and the curvature; decompose the acceleration vector into its tangential and normal components (a ˆT and a ˆN, respectively); * apply vector operations to the position, velocity and acceleration vectors; use the results of these operations to analyze the behavior of the motion of an object traveling through space; * use the decomposition of acceleration into tangential and normal components to draw conclusions about speed, velocity and curvature, and vice versa. DERIVATIVES OF MULTIVARIABLE FUNCTIONS There are several notions of derivative associated with a function of more than one variable. Partial derivatives, which are analogous to derivatives of functions of a single variable, give the rates of change of the function with respect to directions in the domain parallel to the coordinate axes. Directional derivatives also generalize single-variable derivatives, and give the rate of change of the function with respect to any direction in the domain. The chain rule arises from taking derivatives along arbitrary paths in the domain. The differential of a function reflects the change in a function s value over an infinitesimal displacement in the domain. The differential can used to approximate the change in a differentiable function value over a small (rather than infinitesimal) displacement. The tangent plane to a surface at a point is the plane passing through the point that most closely approximates the surface. If the surface is the graph of a differentiable function f, the tangent plane at a point is the graph of the local linearization of f. Local linearizations are linear functions that can be used to approximate differentiable functions. The gradient of a function is a vector field on the domain of the function. The magnitude and direction of the gradient give the maximum rate of change of a function and the direction in which it occurs. Gradients are orthogonal to level sets (curves or surfaces in the domain on which the function value is constant). 9
Applications of derivatives, tangent planes, differentials and the gradient include: finding rates of change of a multivariable function depending on direction or on a path of travel in the domain (derivatives); approximating function values (local linearization); approximating the change in the value of a function (differential); determining the sensitivity of a function to perturbations in the independent variables (differential); finding local maxima and minima and saddle points of a function (critical points and second derivative test); optimization subject to constraint (Lagrange multipliers). For this exam, you are expected to be able to: * perform basic computations: given a multivariable function, be able to compute its partial derivatives, directional derivatives, differential, and local linearizations/tangent planes; given a multivariable function and a parameterized curve, be able to compute the derivative of the function with respect to the parameter via the chain rule; be able to find the general equation for the level curves of a multivariable function; * understand the relationship of the gradient to directional derivatives; find directions in which a function is changing at a specified rate; * recognize whether a diagram represents the level curves of a function of two variables; understand the geometric relationship between the level curves and the gradient of a function; * apply the various types of derivatives, the differential and the local linearization/tanget plane to setting up and solving word problems; * understand the relationship between the tangent plane and local maxima/minima and saddle points; understand the conditions under which the second derivative test will fail to return a conclusive answer about local maxima/minima and saddle points. You will not be asked about Lagrange Multipliers on this exam. MULTIPLE INTEGRALS The (scalar) area element da determines how area is measured in a double integral. The volume element dv determines how volume is measured in a triple integral. Evaluating multiple integrals with continuous integrands is accomplished by sequentially evaluating single integrals (Fubini s theorem). It is often useful to change to a different coordinate system, which requires not only converting the integrand to the new coordinates, but also using the appropriate area or volume element. 10
Applications include: area of planar regions; volume; average value of a function over a region; mass, first and second moments and centers of mass of two- and three-dimensional regions. For this exam, you are expected to be able to: * perform basic computations: set up and evaluate double and triple integrals in Cartesian, polar, cylindrical and spherical coordinates; * find the region of integration corresponding to a given integral; * convert integrals from one coordinate system to another; * compute the mass, center of mass and first and second moments of twoand three-dimensional objects. 11