DefinitionsChapter 14Prof. D.R. HundleyWhitman CollegeSpring 2009DRHundley (WHI) Math 225 Feb 18 2009 1 / 9IntroductionDefinitionThe natural domain (or just domain) of f (x, y) is the set of orderedpairs (x, y) for which f is defined.ExampleFind the domain: f (x, y) =√y +p25 − x2− y2√y ⇒ y ≥ 025 − x2− y2≥ 0 ⇒ x2+ y2≤ 25DRHundley (WHI) Math 225 Feb 18 2009 2 / 9IntroductionDefinitionThe natural domain (or just domain) of f (x, y) is the set of orderedpairs (x, y) for which f is defined.ExampleFind the domain: f (x, y) =√y +p25 − x2− y2√y ⇒ y ≥ 025 − x2− y2≥ 0 ⇒ x2+ y2≤ 25DRHundley (WHI) Math 225 Feb 18 2009 2 / 9IntroductionDefinitionThe natural domain (or just domain) of f (x, y) is the set of orderedpairs (x, y) for which f is defined.ExampleFind the domain: f (x, y) =√y +p25 − x2− y2√y ⇒ y ≥ 025 − x2− y2≥ 0 ⇒ x2+ y2≤ 25DRHundley (WHI) Math 225 Feb 18 2009 2 / 9IntroductionDefinitionThe natural domain (or just domain) of f (x, y) is the set of orderedpairs (x, y) for which f is defined.ExampleFind the domain: f (x, y) =√y +p25 − x2− y2√y ⇒ y ≥ 025 − x2− y2≥ 0 ⇒ x2+ y2≤ 25DRHundley (WHI) Math 225 Feb 18 2009 2 / 9IntroductionDRHundley (WHI) Math 225 Feb 18 2009 3 / 9IntroductionDefinitionThe graph of:f (x, y) = kis a curve in the plane. We think of the expression as implicitly defining yin terms of x.Example: x2+ 3xy − y3= 5DRHundley (WHI) Math 225 Feb 18 2009 4 / 9IntroductionDefinitionThe graph of:f (x, y) = kis a curve in the plane. We think of the expression as implicitly defining yin terms of x.Example: x2+ 3xy − y3= 5DRHundley (WHI) Math 225 Feb 18 2009 4 / 9IntroductionDefinitionThe graph of a function z = f (x, y) is the set of ordered triples (x, y, z)so that z = f (x, y).We visualize the graph in three dimensions.DefinitionThe graph of a function w = f (x, y , z) would have to have fourdimensions (in order to plot four-tuples: (x, y , z, w ), and you could notvisualize the same way we visualize other graphs.We might think of time as one of the dimensions...To visualize three-dimensional graphs, look at the “level curves”:DefinitionThe level curves for a function z = f (x, y ) are curves where k = f (x, y )(and note these are curves in the plane).DRHundley (WHI) Math 225 Feb 18 2009 5 / 9IntroductionDefinitionThe graph of a function z = f (x, y ) is the set of ordered triples (x, y , z)so that z = f (x, y).We visualize the graph in three dimensions.DefinitionThe graph of a function w = f (x, y , z) would have to have fourdimensions (in order to plot four-tuples: (x, y , z, w ), and you could notvisualize the same way we visualize other graphs.We might think of time as one of the dimensions...To visualize three-dimensional graphs, look at the “level curves”:DefinitionThe level curves for a function z = f (x, y ) are curves where k = f (x, y )(and note these are curves in the plane).DRHundley (WHI) Math 225 Feb 18 2009 5 / 9IntroductionDefinitionThe graph of a function z = f (x, y ) is the set of ordered triples (x, y , z)so that z = f (x, y).We visualize the graph in three dimensions.DefinitionThe graph of a function w = f (x, y , z) would have to have fourdimensions (in order to plot four-tuples: (x, y , z, w ), and you could notvisualize the same way we visualize other graphs.We might think of time as one of the dimensions...To visualize three-dimensional graphs, look at the “level curves”:DefinitionThe level curves for a function z = f (x, y ) are curves where k = f (x, y )(and note these are curves in the plane).DRHundley (WHI) Math 225 Feb 18 2009 5 / 9IntroductionThe level curves are curves where the function is constant.In weather maps, these are curves where the pressure is constant.DRHundley (WHI) Math 225 Feb 18 2009 6 / 9IntroductionThe level curves are curves where the function is constant.In weather maps, these are curves where the pressure is constant.DRHundley (WHI) Math 225 Feb 18 2009 6 / 9IntroductionDRHundley (WHI) Math 225 Feb 18 2009 7 / 9IntroductionDRHundley (WHI) Math 225 Feb 18 2009 7 / 9IntroductionReading Level CurvesIf a surface is very steep -Level curves are packed together.Shallow surfaceLevel curves are far apart.Now a mathematical example:DRHundley (WHI) Math 225 Feb 18 2009 8 / 9IntroductionReading Level CurvesIf a surface is very steep -Level curves are packed together.Shallow surfaceLevel curves are far apart.Now a mathematical example:DRHundley (WHI) Math 225 Feb 18 2009 8 / 9IntroductionReading Level CurvesIf a surface is very steep -Level curves are packed together.Shallow surfaceLevel curves are far apart.Now a mathematical example:DRHundley (WHI) Math 225 Feb 18 2009 8 / 9IntroductionReading Level CurvesIf a surface is very steep -Level curves are packed together.Shallow surfaceLevel curves are far apart.Now a mathematical example:DRHundley (WHI) Math 225 Feb 18 2009 8 / 9IntroductionReading Level CurvesIf a surface is very steep -Level curves are packed together.Shallow surfaceLevel curves are far apart.Now a mathematical example:DRHundley (WHI) Math 225 Feb 18 2009 8 / 9IntroductionExample: Plot the function (x2+ 3y2)e−x2−y2by looking at it in 3-d andwith the level curves:DRHundley (WHI) Math 225 Feb 18 2009 9 /