Introduction to Mathematical Physics. Methods and Concepts by Wong C.W.

By Wong C.W.

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By Wong C.W.

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The chain rule of differential calculus, Eq. 54), now requires the replacements B(A · C) = (C · A)B → B(∇ · C) + (C · ∇)B −C(A · B) = −(B · A)C → −C(∇ · B) − (B · ∇)C, with two terms appearing for each term of the BAC rule. Thus Eq. 60) obtains. We shall refer to this procedure as the operator form of the BAC rule. 1 Derive the following identities: (a) ∇ · (A × B) = B · (∇ × A) − A · (∇ × B); (b) ∇ × (∇ × A) = ∇(∇ · A) − ∇2 A; (c) ∇(A · B) = (B · ∇)A + (A · ∇)B + B × (∇ × A) + A × (∇ × B). 2 If A(r) is irrotational, show that A × r is solenoidal.

Some of the most interesting vector fields in physics satisfy the property that the results of such vector differential operations vanish everywhere in space. Such special fields are given special names: If ∇ · V(r) = 0, V is said to be solenoidal (or divergence-free). If ∇ × V(r) = 0, V is said to be irrotational. 3 Show that the gradient of any scalar field Φ(r) is irrotational and that the curl of any vector field V(r) is solenoidal. i j k ∇ × (∇Φ) = ∂ ∂x ∂ ∂x ∂ ∂y ∂ ∂y ∂ ∂z ∂ ∂z Φ(x, y, z) = 0 because there are two identical rows in the determinant.

4 Operator identity Gauss’s theorem, Eq. 85), can be applied to the vector field V(r) = ei Φ(r) to give dσi Φ(r) = Ω S dτ ∂ Φ(r). ∂xi Since this is true for any i, it follows that dσΦ(r) = S = dσi Φ(r) ei S i Ω dτ∇Φ(r). 93) for operations on any field in space. It gives rise to other integral theorems such as the following: dσ × A(r) = S dσ · (u∇v − v∇u) = S = Ω Ω Ω dτ∇ × A(r), dτ∇ · (u∇v − v∇u) dτ(u∇2 v − v∇2 u). The last identity is called Green’s theorem. 2 Show that dσ · r = r2 dτ 1 . 94) r 1 = 2, 2 r r a result that can be demonstrated readily in rectangular coordinates.

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