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l_notes10 - p is a function of volume V 1 2 ± and internal...

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1 Lecture X Linear Approximation Chain Rule Linear Approximation; Gradient We say that a function has a linear approximation on a domain D if it has a linear approximation at any point P D . Theorem 1 If f has a linear approximation on a domain D , then f is contin- uous on D . Theorem 2 If f has continuous partial derivatives on D , then f has linear approximation on D . Defnition 1 Let f be a function with partial derivatives at P . The vector ∂f ˆ ∂f ˆ G P = ˆ i + ∂f j + k ∂x ∂y ∂z is called the gradient at P and is usually denoted by P . f | Theorem 3 (Gradient Theorem) Let f have domain D in E 3 and let P be a point in D . If ˆ u is a Fxed unit vector, then df = ∂f ˆ i + ∂f ∂f ˆ j + ˆ k = ˆ u f P · | ds u,P ∂x P ∂y P ∂z P ˆ Chain Rule Physical variables, quantities that can be measured in a given physical system, are related in various ways. For example, pressure p is a function of volume V and temperature T , p = h ( V, T ). Also, T depends on V and internal energy U , T = t ( V, U ). From these to equalities, we get that

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Unformatted text preview: p is a function of volume V 1 2 ± and internal energy U , p = h ( V, T ) = h ( V, t ( V, U )) = g ( V, U ). So to fnd the partial derivative oF p with respect to V , we have to specify the Function we = ∂h are reFerring to: h or g . We write ( ∂p = ∂g and ( ∂p . Below we ∂V ) U ∂V ∂V ) T ∂V will see how such partial derivatives are related. By ( ∂f ∂x ) y we mean the partial derivative oF f with respect to x when f is seen as a Function oF variables x and y . Theorem 4 (Chain Rule) Let w, u, v, x, y be physical variables such that there exist diﬀerentiable functions f, g, h such that w = f ( u, v ) , u = g ( x, y ) and v = h ( x, y ) . Then ± ∂w = ∂f ∂g + ∂f ∂h ∂x y ∂u ∂x ∂v ∂x ± ∂w = ∂f ∂g + ∂f ∂h ∂y x ∂u ∂y ∂v ∂y IF we take w = f ( u, v ), u = g ( x, y ), and v = h ( x ), then applying the Chain Rule we get ∂w ∂f ∂g = , ∂y x ∂u ∂y ∂h since = 0. ∂y 2...
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l_notes10 - p is a function of volume V 1 2 ± and internal...

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