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LanguageExamplesBridging the Gap between Mathematicsand the Physical SciencesTevian Dray & Corinne A. ManogueDepartments of Mathematics & PhysicsOregon State Universityhttp://www.math.oregonstate.edu/~tevianhttp://physics.oregonstate.edu/~corinneTevian Dray & Corinne A. Manogue Bridging the Gap between Mathematics and the Physical Scienceshttp://www.math.oregonstate.edu/~tevianhttp://physics.oregonstate.edu/~corinneLanguageExamplesMath vs. PhysicsFunctionsContentMathematics vs. PhysicsTevian Dray & Corinne A. Manogue Bridging the Gap between Mathematics and the Physical SciencesLanguageExamplesMath vs. PhysicsFunctionsContentMathematics vs. PhysicsTevian Dray & Corinne A. Manogue Bridging the Gap between Mathematics and the Physical SciencesLanguageExamplesMath vs. PhysicsFunctionsContentMathematics vs. PhysicsTevian Dray & Corinne A. Manogue Bridging the Gap between Mathematics and the Physical SciencesLanguageExamplesMath vs. PhysicsFunctionsContentWhat are Functions?Suppose the temperature on a rectangular slab of metal is given byT (x , y) = k(x2 + y2)where k is a constant. What is T (r , )?Tevian Dray & Corinne A. Manogue Bridging the Gap between Mathematics and the Physical SciencesLanguageExamplesMath vs. PhysicsFunctionsContentWhat are Functions?Suppose the temperature on a rectangular slab of metal is given byT (x , y) = k(x2 + y2)where k is a constant. What is T (r , )?A: T (r , ) = kr2B: T (r , ) = k(r2 + 2)yrxTevian Dray & Corinne A. Manogue Bridging the Gap between Mathematics and the Physical SciencesLanguageExamplesMath vs. PhysicsFunctionsContentWhat are Functions?MATHT = f (x , y) = k(x2 + y2)T = g(r , ) = kr2PHYSICST = T (x , y) = k(x2 + y2)T = T (r , ) = kr2Tevian Dray & Corinne A. Manogue Bridging the Gap between Mathematics and the Physical SciencesLanguageExamplesMath vs. PhysicsFunctionsContentWhat are Functions?MATHT = f (x , y) = k(x2 + y2)T = g(r , ) = kr2PHYSICST = T (x , y) = k(x2 + y2)T = T (r , ) = kr2Two disciplines separated by a common language...Tevian Dray & Corinne A. Manogue Bridging the Gap between Mathematics and the Physical SciencesLanguageExamplesMath vs. PhysicsFunctionsContentMathematics vs. PhysicsPhysics is about things.Physicists cant change the problem.Mathematicians do algebra.Physicists do geometry.Tevian Dray & Corinne A. Manogue Bridging the Gap between Mathematics and the Physical SciencesLanguageExamplesMath vs. PhysicsFunctionsContentContent MattersWhat do we teach?What do we leave out?Do we teach concepts, facts, problem-solving, ...?What order do we teach it in?How do the pieces contribute to our overall goals?Tevian Dray & Corinne A. Manogue Bridging the Gap between Mathematics and the Physical SciencesLanguageExamplesMath vs. PhysicsFunctionsContentEarly Mathematics ContentWhat is the role of the number 2?2 sin xsin 2xsin(2 + x)Tevian Dray & Corinne A. Manogue Bridging the Gap between Mathematics and the Physical SciencesLanguageExamplesMath vs. PhysicsFunctionsContentPhysics Application: WavesA sin(kx t)Dependence on two variablesMinus signParameters rather than constantsFunny Greek letterPhysics contentTevian Dray & Corinne A. Manogue Bridging the Gap between Mathematics and the Physical SciencesLanguageExamplesDot ProductDictionaryGradientRecord on your small white board somethingthat you know about the dot product.Tevian Dray & Corinne A. Manogue Bridging the Gap between Mathematics and the Physical SciencesLanguageExamplesDot ProductDictionaryGradientRecord on your small white board somethingthat you know about the dot product.Geometry:~u ~v = |~u||~v| cos Algebra:~u ~v = uxvx + uyvyTevian Dray & Corinne A. Manogue Bridging the Gap between Mathematics and the Physical SciencesLanguageExamplesDot ProductDictionaryGradientSmall Whiteboard QuestionsAllow contrast between multiple representations.Elicit common misconceptions.Foster sensemaking discussions.Allow the instructor to see if everyone is on the same page.Encourage quiet members of the class to participate.Keep everyone engaged and awake.Tevian Dray & Corinne A. Manogue Bridging the Gap between Mathematics and the Physical SciencesLanguageExamplesDot ProductDictionaryGradientFind the angle between the diagonal of acube and the diagonal of one of its faces.Tevian Dray & Corinne A. Manogue Bridging the Gap between Mathematics and the Physical SciencesLanguageExamplesDot ProductDictionaryGradientFind the angle between the diagonal of acube and the diagonal of one of its faces.Algebra:~u = + + k~v = + k= ~u ~v = 2Geometry:~u ~v = |~u||~v| cos =32 cos Tevian Dray & Corinne A. Manogue Bridging the Gap between Mathematics and the Physical SciencesLanguageExamplesDot ProductDictionaryGradientFind the angle between the diagonal of acube and the diagonal of one of its faces.Algebra:~u = + + k~v = + k= ~u ~v = 2Geometry:~u ~v = |~u||~v| cos =32 cos Need both!Tevian Dray & Corinne A. Manogue Bridging the Gap between Mathematics and the Physical SciencesLanguageExamplesDot ProductDictionaryGradientKerry Browne (Ph.D. 2002)Tevian Dray & Corinne A. Manogue Bridging the Gap between Mathematics and the Physical SciencesLanguageExamplesDot ProductDictionaryGradientGradientGradientThe gradient of a function is a vector field that points in thedirection in which the function increases most rapidly, and whosemagnitude is the amount of that increase.Tevian Dray & Corinne A. Manogue Bridging the Gap between Mathematics and the Physical SciencesLanguageExamplesDot ProductDictionaryGradientThe HillSuppose you are standing on ahill. You have a topographic map,which uses rectangular coordinates(x , y) measured in miles. Yourglobal positioning system says yourpresent location is at one of thepoints shown. Your guidebook tellsyou that the height h of the hill infeet above sea level is given byh = a bx2 cy2where a = 5000 ft, b = 30 ftmi2,and c = 10 ftmi2. -10-50510y-6 -4 -2 0 2 4 6xTevian Dray & Corinne A. Manogue Bridging the Gap between Mathematics and the Physical SciencesLanguageMath vs. PhysicsFunctionsContentExamplesDot ProductDictionaryGradient

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