Michael Corral: Vector Calculus

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100 CHAPTER 2. FUNCTIONS OF SEVERAL VARIABLES Forexample,inExample2.25weshowedthattheconstrainedoptimizationproblem Maximize : f(x,y)= xy given : g(x,y)=2x+2y=20 had the solution (x,y)=(5,5), and thatλ= x/2=y/2. Thus,λ=2.5. In a similar fashion we could show that the constrained optimization problem Maximize : f(x,y)= xy given : g(x,y)=2x+2y=21 hasthesolution(x,y)=(5.25,5.25). Soweseethatthevalueof f(x,y)attheconstrained maximum increased from f(5,5)=25 to f(5.25,5.25)=27.5625, i.e. it increased by 2.5625 when we increased the value of c in the constraint equation g(x,y)=c from c=20 to c=21. Notice thatλ=2.5 is close to 2.5625, that is, λ≈∆f=f(new max. pt)− f(old max. pt). Finally, note that solving the equation∇f(x,y)=λ∇g(x,y) means having to solve a system of two (possibly nonlinear) equations in three unknowns, which as we have seen before, may not be possible to do. And the 3-variable case can get even more complicated. All of this somewhat restricts the usefulness of Lagrange’s method to relatively simple functions. Luckily there are many numerical methods for solving constrained optimization problems, though we will not discuss them here. 13 A ☛ ✟ ✡Exercises ✠ 1. Find the constrained maxima and minima of f(x,y)=2x+y given that x 2 +y 2 = 4. 2. Find the constrained maxima and minima of f(x,y)= xy given that x 2 +3y 2 = 6. 3. Find the points on the circle x 2 +y 2 = 100 which are closest to and farthest from the point (2,3). B 4. Find the constrained maxima and minima of f(x,y,z)= x+y 2 +2z given that 4x 2 + 9y 2 −36z 2 = 36. 5. Find the volume of the largest rectangular parallelepiped that can be inscribed in the ellipsoid x 2 a 2+y2 b 2+z2 c2= 1. 13 See BAZARAA, SHERALI and SHETTY.
3 Multiple Integrals 3.1 Double Integrals In single-variable calculus, differentiation and integration are thought of as inverse operations. For instance, to integrate a function f(x) it is necessary to find the antiderivative of f, that is, another function F(x) whose derivative is f(x). Is there a similar way of defining integration of real-valued functions of two or more variables? The answer is yes, as we will see shortly. Recall also that the definite integral of a nonnegative function f(x)≥0 represented the area “under” the curve y= f(x). As we will now see, the double integral of a nonnegative real-valued function f(x,y)≥0 represents the volume “under” the surface z= f(x,y). Let f(x,y)beacontinuousfunctionsuchthat f(x,y)≥0forall(x,y)ontherectangle R={(x,y) : a≤ x≤b, c≤y≤d} in 2 . We will often write this as R=[a,b]×[c,d]. For any number x∗ in the interval [a,b], slice the surface z= f(x,y) with the plane x= x∗ parallel to the yz-plane. Then the trace of the surface in that plane is the curve f(x∗,y), where x∗ is fixed and only y varies. The area A under that curve (i.e. the area of the region between the curve and the xy-plane) as y varies over the interval [c,d] then depends only on the value of x∗. So using the variable x instead of x∗, let A(x) be that area (see Figure 3.1.1). z z= f(x,y) x a c 0 A(x) d y b x R Figure 3.1.1 The area A(x) varies with x Then A(x)= ∫ d f(x,y)dy since we are treating x as fixed, and only y varies. This c makes sense since for a fixed x the function f(x,y) is a continuous function of y over the interval [c,d], so we know that the area under the curve is the definite integral. 101
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About the author: Michael Corral is
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2 CHAPTER 1. VECTORS IN EUCLIDEAN S
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150 CHAPTER 4. LINE AND SURFACE INT
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188 Bibliography Pogorelov, A.V., A
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190 Appendix A: Answers and Hints t
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Appendix B We will prove the right-
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194 Appendix B: Proof of the Right-
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Appendix C 3D Graphing with Gnuplot
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198 Appendix C: 3D Graphing with Gn
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200 Appendix C: 3D Graphing with Gn
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202 GNU Free Documentation License
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Index Symbols D....................
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212 Index iterated integral .......
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Michael Corral: Vector Calculus

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