Mixed Derivative Theorem

Friday, 10 August 2018

3:52 PM

If 𝐹(𝑥,𝑦) is a continuous function of two variables and all of its first and second order partial derivatives are continuous then ﷐﷐δ﷮2﷯𝐹﷮𝛿𝑥𝛿𝑦﷯=﷐﷐𝛿﷮2﷯𝐹﷮𝛿𝑦𝛿𝑥﷯ Tangent Planes and Surface Normal 𝑦=𝑓﷐𝑥﷯ 𝑦−﷐𝑦﷮0﷯=﷐𝑓﷮′﷯﷐﷐𝑥﷮0﷯﷯﷐𝑥−﷐𝑥﷮0﷯﷯ Normal = −﷐1﷮﷐𝑓﷮′﷯﷐﷐𝑥﷮0﷯﷯﷯ 𝑥−﷐𝑥﷮0﷯=﷐𝑓﷮′﷯﷐﷐𝑥﷮0﷯﷯﷐𝜆﷯ 𝑖𝑓 𝜆=﷐𝑦﷮0﷯−𝑦 then, ﷐﷐𝑥﷮𝑦﷯﷯=﷐﷐﷐𝑥﷮0﷯﷮﷐𝑦﷮0﷯﷯﷯+𝜆﷐﷐𝑓′(﷐𝑥﷮0﷯)﷮−1﷯﷯ So the normal vector to the curve 𝑦=𝑓(𝑥) at (﷐𝑥﷮0﷯,﷐𝑦﷮0﷯) is ﷐﷐𝑓′(﷐𝑥﷮0﷯)﷮−1﷯﷯ 3D Normal Vector to the Surface at ﷐﷐𝑥﷮0﷯,﷐𝑦﷮0﷯,﷐𝑧﷮0﷯﷯ ﷐﷐﷐𝐹﷮𝑥﷯(﷐𝑥﷮0﷯,﷐𝑦﷮0﷯)﷮﷐𝐹﷮𝑦﷯(﷐𝑥﷮0﷯,﷐𝑦﷮0﷯)﷮−1﷯﷯ Find the equation of the tangent ﷐﷐𝑥−﷐𝑥﷮0﷯﷮𝑦−﷐𝑦﷮0﷯﷮𝑧−﷐𝑧﷮0﷯﷯﷯ ﷐ ﷯ ﷐﷐﷐𝐹﷮𝑥﷯﷐﷐𝑥﷮0﷯,﷐𝑦﷮0﷯﷯﷮﷐𝐹﷮𝑦﷯﷐﷐𝑥﷮0﷯,﷐𝑦﷮0﷯﷯﷮−1﷯﷯=0 ** Perpendicular to the normal is the tangent, so we can use the dot product!! Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings 

 

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