Q. 19 聽Give a formula for a normal ve... [FREE SOLUTION]

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Chapter 14: Q. 19 (page 1119)

Give a formula for a normal vector to the surface S determined by x = f(y, z), where f(y, z) is a function with continuous partial derivatives.

Short Answer

Expert verified

$Therefor the required normal vector is = i 鈭� f_{y} j 鈭� f_{z} k .$

Step by step solution

Step 1. Given

f(y, z) is a function with continuous partial derivatives.

Step 2. Find ry and rz

$B y a b o v e s t e p s, t h e s u r f a c e S i s p a r a m e t r i z e d b y a s f o l l o w s r (y, z) = <f (y, z), y, z> T h e n \begin{aligned} r_{y} = \frac{鈭�}{鈭� y} r (y, z) \\ = \frac{鈭�}{鈭� y} 鉄� f (y, z), y, z 鉄� \\ = <\frac{鈭� f}{鈭� y}, 1, 0> \\ = <f_{y}, 1, 0> \end{aligned} a n d \begin{aligned} r_{z} = \frac{鈭�}{鈭� z} r (y, z) \\ = \frac{鈭�}{鈭� z} 鉄� f (y, z), y, z 鉄� \\ = <\frac{鈭� f}{鈭� z}, 0, 1> \\ = <f_{z}, 0, 1> \end{aligned}$

Step 3. Find n=rn×ry

$\begin{aligned} n = r_{y} 脳 r_{z} \\ = <f_{y}, 1, 0> 脳 <f_{z}, 0, 1> \\ = |\begin{array}{lll} i & j & k \\ f_{y} & 1 & 0 \\ f_{z} & 0 & 1 \end{array}| \\ = [1 鈰� 1 鈭� 0 鈰� 0] i 鈭� [f_{y} 鈰� 1 鈭� 0 鈰� f_{z}] j + [f_{y} 鈰� 0 鈭� 1 鈰� f_{z}] k \\ = i 鈭� f_{y} j 鈭� f_{z} k . \end{aligned} Therefor the required normal vector is = i 鈭� f_{y} j 鈭� f_{z} k .$

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91影视

Give a formula for a normal vector to the surface S determined by x = f(y, z), where f(y, z) is a function with continuous partial derivatives.

Short Answer

Step by step solution

Step 1. Given

Step 2. Find ry and rz

Step 3. Find n=rn×ry

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