Engineering Calculus Notes 265

Engineering Calculus Notes 265 - cannot be done if the...

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3.4. LEVEL CURVES 253 Challenge problem: 9. Show that the if f ( x,y ) = g ( ax + by ) where g ( t ) is a diFerentiable function of one variable, then for every point ( x,y ) in the plane with equation ax + by = c (for some constant c ), −→ f is perpendicular to this plane. 10. (a) Show that if f ( x,y ) is a function whose value depends only on the product xy then x ∂f ∂x = y ∂f ∂y . (b) Is the converse true? That is, suppose f ( x,y ) is a function satisfying the condition above on its partials. Can it be expressed as a function of the product f ( x,y ) = g ( xy ) for some real-valued function g ( t ) of a real variable? ( Hint: ±irst, consider two points in the same quadrant, and join them with a path on which the product xy is constant. Not that this
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Unformatted text preview: cannot be done if the points are in diFerent quadrants.) 11. Adapt the proof of Theorem 3.3.4 given in this section for functions of two variables to get a proof for functions of three variables. 3.4 Level Curves A level set of a function f is any subset of its domain of the form L ( f,c ) := { −→ x | f ( −→ x ) = c } where c ∈ R is some constant. This is nothing other than the solution set of the equation in two or three variables f ( x,y ) = c or f ( x,y,z ) = c. ±or a function of two variables, we expect this set to be a curve in the plane and for three variables we expect a surface in space....
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