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Engineering Calculus Notes 276

Engineering Calculus Notes 276 - 264 CHAPTER 3 REAL-VALUED...

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264 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION or ∂f ∂y φ ( x ) = ∂f ∂x φ ( x ) = ∂f/∂x ∂f/∂y as required. A mnemonic device to remember which partial goes on top of this fraction and which goes on the bottom is to write Equation ( 3.18 ) formally as dy dx = dy dz dz dx –that is, we formally (and unjustifiably) “cancel” the dz terms of the two “fractions”. (Of course, we have to remember separately that we need the minus sign up front.) Equation ( 3.18 ) can also be interpreted as saying that a vector tangent to the level curve has slope φ ( x ) = bracketleftbigg ∂f ∂x ( x,φ ( x )) bracketrightbiggslashbiggbracketleftbigg ∂f ∂y ( x,φ ( x )) bracketrightbigg , which means that it is perpendicular to −→ f ( x,φ ( x )). Of course, this could also be established using the Chain Rule (Exercise
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