Math 275, section 002 (Ultman) Fall 2011 FINAL EXAM REVIEW The final exam will be held on Wednesday 14 December from 10:30am 12:30pm in our regular classroom. You will be allowed both sides of an 8.5 11 sheet 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 40 80%) will address material covered in class since the third exam, corresponding to chapter 15 in the text. The remainder will be material from earlier in the semster. This review includes suggested review problems. In addition, I recommend looking at the Questions to Guide Your Review at the end of each chapter. 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, quizzes, exams, etc. CURRENT MATERIAL: Vector Calculus. Suggested review problems: sec. 15.1 # 29, 31, 34 sec. 15.2 # 7, 23, 26, 53 sec. 15.3 # 26, 32, 35, 36, 38 sec. 15.5 # 17, 19, 21, 23, 25 sec. 15.6 # 15, 22, 23, 24, 27, 28, 37, 39 sec. 15.4, 15.7, 15.8: problems in Assignment 14.
LINE INTEGRALS Key Ideas: 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. Given a smooth parameterization for a curve, ds gives a way to measure distance along the curve. Applications of scalar line integrals include: length of curves; mass, first and second moments and centers of mass of wires. The component of a field F acting along an infinitesimal section of a curve is given by the dot product of F d r = F ˆT ds, where ˆT is the unit tangent vector to the curve. Vector line integrals add up these scaled projections along the curve. Applications of vector line integrals include: work ( F represents force which acts over the infinitesimal displacement d r) and circulation ( F is a velocity field). 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 around closed curves equal zero; F is a gradient field. If a vector field is defined and continuously differentiable on a simply connected domain, then the curl test can be used to determine whether the field is conservative. The Fundamental Theorem of Line Integrals states that for a conservative field F and piecewise smooth curve C, the integral C F d r can be computed by evaluating a potential function at the endpoints and subtracting the value at the initial point from the value at the terminal point (cf. the Fundamental Theorem of Calculus). One method for finding the potential function of a conservative field is to integrate the component functions with respect to the appropriate variables and compare the results. 2
SURFACE AND FLUX INTEGRALS Key Ideas: Surface integrals generalize double integrals over a region in the xy-plane to integrals defined on (possibly) non-planar surfaces. Evaluation of surface integrals requires either a smooth parameterization of the surface or some type of geometric insight. 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. Applications of surface integrals include: surface area; mass, first and second moments and centers of mass of shells. The component of a field F passing through an infinitesimal region of a surface is given by the dot product F d A = F ˆn da, where ˆn is a field of unit vectors normal to the surface. Given a smooth parameterization r(u, v), the vector area element can be computed by d A = ± d r u d r v. The flux of the field F through the oriented surface S is given by the integral S F d A. Key Ideas: STOKES THEOREM AND THE DIVERGENCE THEOREM 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 pointing 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; 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 can be determined by integrating the divergence of the field over the solid region enclosed by the surface. 3
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. In fact, we could throw in the Fundamental theorem of calculus as well; the relationship between the FTC and the FTLI is similar to that between Green s and Stokes theorems. 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: 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. SUMMARY OF THE THREE BIG THEOREMS OF VECTOR CALCULUS 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 4
MATERIAL PRECEDING THE THIRD EXAM For a more detailed review of this material, see the reviews for the previous exams. Suggested review problems: ch. 11: sec. 11.2 # 49 sec. 11.3 # 4, 16, 24, 41, 44 sec. 11.4 # 23, 27, 28, 29, 31, 47 sec. 11.5 # 3, 6, 9, 21, 23, 26 ch. 12: sec. 12.1 # 7, 8, 15, 19, 23 25 sec. 12.3 # 9, 18, 19 sec. 12.4 # 7, 9, 17 sec. 12.5 # 1, 3, 17, 18 ch. 13: sec. 13.1 # 13, 15, 31 37, 42 sec. 13.3 # 39, 56, 90 sec. 13.4 # 35, 41, 42, 47, 49 sec. 13.5 # 11, 17, 21, 23, 30, 34, 35 sec. 13.6 # 23, 50, 51 sec. 13.7 # 41, 43, 44, 46, 47 sec. 13.8 # 9, 11 ch. 14: sec. 14.2 # 23, 45, 57, 77, 78, 80 sec. 14.3 # 15, 18, 20, 21 sec. 14.4 # 21, 25, 34, 35, 37 sec. 14.5 # 21, 27, 31, 41 sec. 14.6 # 9, 11 sec. 14.7 # 39 41, 67 sec. 14.8 # 17, 20. VECTORS Vectors are object described by a scalar component (magnitude) and a direction. Vectors can be added and multiplied by scalars. Vector addition and scalar multiplication can be used in finding the parameterization of a line. 5
The dot product, which is a scalar, can be thought of as a symmetric vector projection, v w = v w cos θ. The dot product is used to define orthogonality of vectors. Applications of the dot product include: distance between points; magnitude of vectors; equations of spheres; angles between lines or vectors; equations of lines in R 2 and planes in R 3 ; computing work. 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. 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. 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. 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 zero. 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: projectile motion (integration); motion in space (velocity, acceleration, curvature, the { ˆT, ˆN, ˆB} frame); length of curves. 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 6
the rate of change of the function with respect to any direction in the domain. The chain rule arises from taking derivatives along arbitrary parameterized curves in the domain. The differential reflects the change in a function s value over an infinitesimal displacement in the domain; compare this to the derivatives, which give various rates of change of the function. 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, the tangent plane at a point is the graph of linear function, called the local linearization. Local linearizations can be used to approximate differentiable functions. All information about partial and directional derivatives, the chain rule, the differential and the tangent plane are encoded by the gradient of the function. 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). Derivatives and differentials are encoded by the Master Formula df = f d r. Applications of derivatives, tangent planes, differentials and the gradient include: information about rates of change of a function and changes in function value; approximating function values; determining the sensitivity of a function to perturbations in the independent variables; finding maxima and minima of a function (critical points and second derivative test); optimization subject to constraint (Lagrange multipliers). 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 7
is accomplished by sequentially evaluating single integrals (Fubini s theorem). It is often useful to change to a different coordinate system, which requires using the appropriate area or volume element (see the section on change of variables ). Applications include: planar area; volume; average value of a function over a region; mass, first and second moments and centers of mass of twoand three-dimensional regions. CHAGE OF VARIABLES Changing variables in a double or triple integral is carried out by reparameterizing the region over which the integral is being evaluated. Before integrating, one must find the appropriate area or volume element corresponding to the re-parameterization (note that this re-parameteriaztion is also called a coordinate transform). This can be accomplished in at least three different ways: using geometric insight (as was done in the case of polar, cylindrical and spherical coordinates); if the region is planar, computing the magnitude of the vector area element d A; finding the magnitude of the Jacobian of the coordinate transform. In the plane, given a smooth parameterization r(u, v) corresponding to the coordinate transform x = x(u, v), y = y(u, v), the scalar area element da and the magnitude of the Jacobian J(u, v) are related as follows: da = d r u d r v = J(u, v) du dv. In R 3, given a smooth parameterization r(u, v, w) corresponding to the coordinate transform x = x(u, v, w), y = y(u, v, w), z = z(u, v, w): dv = (d r u d r v ) d r w = J(u, v, w) du dv dw. 8