Level Curves and Profiles

Friday, 10 August 2018

3:17 PM

A level curve of a function of two variables 𝐹:﷐ℝ﷮2﷯→ℝ is a curve in the x-y plane, defined by 𝐹﷐𝑥,𝑦﷯=𝐶 They can be obtained by intersecting the graph of F with the horizontal plane 𝑧=𝐶, and then projecting this intersection onto the x-y plane 𝑙𝑒𝑡 𝑧=𝐹﷐𝑥,𝑦﷯ We can obtain each level curve by setting 𝑧=𝐶 for some 𝑟 that we decide 𝑧 Level Curve z=0 ﷐𝑥﷮2﷯+﷐𝑦﷮2﷯=0 z=1 ﷐𝑥﷮2﷯+﷐𝑦﷮2﷯=1 z=4 ﷐𝑥﷮2﷯+﷐𝑦﷮2﷯=4 Sketch the level curves and profiles of 𝐹﷐𝑥,𝑦﷯=﷐𝑥﷮2﷯+﷐𝑦﷮2﷯ Sketch the y-z profile 𝑙𝑒𝑡 𝑥=0 𝑧=﷐𝑥﷮2﷯+﷐𝑦﷮2﷯=0+﷐𝑦﷮2﷯ 𝑧=﷐𝑦﷮2﷯ Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings        
A surface in ﷐𝑅﷮3﷯ is described by the equation ﷐𝑧﷮2﷯=1−﷐𝑥﷮2﷯−﷐𝑦﷮2﷯ Sketch some level curves and hence sketch the profile in R Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings                     Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings

 

 

 

A surface in  is described by the equation

 

Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings                      Ink Drawings Ink Drawings Ink Drawings  
Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings   

 

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