nil asked in 科學數學 · 1 decade ago

# 關於向量分析名詞的定義(20點)

conservative,vector field,divergerce,gradient,curl,potartial function,flux

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• linch
Lv 7
1 decade ago
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以下都是指三度空間

A vector field F on a domain in space is a function that assigns a vector to each point in the domain.

Like: F(x,y,z) = M(x,y,z) i + N(x,y,z) j + P(x,y,z) k

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F is conservative in D ==>∫F•dr (是一個線積分) is path independent in D (線積分與路徑無關)

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The gradient vector of f(x,y,z) is the vector

▽f(x,y,z) = f_x(x,y,z) i + f_y(x,y,z) j + f_z(x,y,z) k

f_x(x,y,z) 表示 f 對 x 偏微

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If F is a field defined on D and F(x,y,z) = ▽f(x,y,z) for some function f on D, then f is called a potartial function for F.

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The divergence of a vector field F(x,y,z) = M(x,y,z) i + N(x,y,z) j + P(x,y,z) k is the scaler function

div F = ▽•F = M_x(x,y,z) + N_y(x,y,z) + P_z(x,y,z)

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The flux of a vector field F across an oriented surface S in the direction of n (normal vector) is given by the formula

Flux = ∫∫_S F•n dσ

∫∫_S F•ndσ 表示 F•n 在 S 上做重積分

2009-01-11 02:26:36 補充：

漏了一個 Curl

F(x,y,z) = M(x,y,z) i + N(x,y,z) j + P(x,y,z) k

Curl F = ▽╳F = (P_y(x,y,z) - N_z(x,y,z)) i + (M_z(x,y,z) - P_x(x,y,z)) j + (N_x(x,y,z) - M_y(x,y,z)) k

2009-01-12 11:43:41 補充：

你要的不是"定義"嗎？

那些都是定義啊！

怎麼又變成解釋定義了？

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