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MDcalc_Diff_3_4

# MDcalc_Diff_3_4 - Multidimensional Calculus Differential...

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Multidimensional Calculus: Differential Theory Chapter & Page: 9–13 9.3 The Classic Gradient, Divergence and Curl Basic Definitions in Euclidean Space For expediency, we will first define the classical differential operators for scalar and vector fields in a Euclidean space E using the “del operator”. Moreover, we will assume that our coordinate system { x 1 , x 2 ,..., x N } is Cartesian . The del operator is the “vector differential operator” given in our Cartesian system by −→ = = N summationdisplay k = 1 x k e k = N summationdisplay k = 1 e k x k = N summationdisplay k = 1 x x k x k . For any sufficiently differentiable scalar field Ψ and vector field F with corresponding coordi- nate formulas Ψ( x ) = ψ ( x 1 , x 2 ,..., x N ) and F ( x ) = N summationdisplay k = 1 F k ( x 1 , x 2 ,..., x N ) e k , we define the gradient of Ψ = grad (Ψ) = Ψ , the divergence of F = div ( F ) = · F , and the curl of F = curl ( F ) = × F by the coordinate formulas grad (Ψ) = Ψ = N summationdisplay k = 1 e k x k ψ ( x 1 , x 2 ,..., x N ) = N summationdisplay k = 1 e k ∂ψ x k = N summationdisplay k = 1 ∂ψ x k e k , div ( F ) = · F = bracketleftBigg N summationdisplay k = 1 x k e k bracketrightBigg · N summationdisplay j = 1 F j ( x 1 , x 2 ,..., x N ) e j = N summationdisplay k = 1 F k x k , and curl ( F ) = × F = bracketleftBigg N summationdisplay k = 1 x k e k bracketrightBigg × N summationdisplay j = 1 F j ( x 1 , x 2 ,..., x N ) e j = vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle e 1 e 2 e 3 x 1 x 2 x 3 F 1 F 2 F 3 vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle = bracketleftbigg F 3 x 2 F 2 x 3 bracketrightbigg e 1 bracketleftbigg F 3 x 1 F 1 x 3 bracketrightbigg e 2 + bracketleftbigg F 2 x 1 F 1 x 2 bracketrightbigg e 3 . Both grad (Ψ) and div ( F ) can be defined assuming our space is of any dimension. However, curl ( F ) requires that our space be three-dimensional (in which case,we usually use the traditional { ( x , y , z ) } and { i , j , k } notation). version: 10/31/2011

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Multidimensional Calculus: Differential Theory Chapter & Page: 9–14 Observe that grad (Ψ) and curl ( F ) are vector fields, while div ( F ) is a scalar field. One thing not obvious from the above definitions is why anyone would be interested in these things. There are good reasons arising from both basic mathematics and from physics. The “geometric/physical significance” of grad (Ψ) will be discussed in a little bit. The importance of the divergence and curl of a vector field will be discussed later, in conjunction with some classic integral theorems. In anticipation of that discussion, let me introduce some terminology that you may encounter in homework: F is solenoidal” ⇐⇒ F is divergence free” ⇐⇒ · F = 0 and F is irrotational” ⇐⇒ F is curl free” ⇐⇒ × F = 0 .
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MDcalc_Diff_3_4 - Multidimensional Calculus Differential...

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