Q. 16 Use the curl form of Green鈥檚 T... [FREE SOLUTION]

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Chapter 14: Q. 16 (page 1132)

Use the curl form of Green鈥檚 Theorem to write the line integral of F(x, y) about the unit circle as a double integral. Do not evaluate the integral.

Short Answer

Expert verified

$The line integral as double integral is - 鈭� {鈭�}_{R} (sinx siny + x^{2} e^{y}) dA$

Step by step solution

Step 1. Given information is:

$F (x, y) = x^{2} e^{y} i + \cos (x) \sin (y) j$

Step 2. Line integral of f(x,y)

$The curl of a vector field F (x, y) = F_{1} (x, y) i + F_{2} (x, y) j in {鈩�}^{2} is defined as follows : curl F = (\frac{鈭� F_{2}}{鈭� x} - \frac{鈭� F_{1}}{鈭� y}) k For the vector field, F (x, y) = x^{2} e^{y} i + \cos (x) \sin (y) j F_{1} (x, y) = x^{2} e^{y} F_{2} (x, y) = \cos (x) \sin (y) T h e n, c u r l F = (\frac{鈭� F_{2}}{鈭� x} - \frac{鈭� F_{1}}{鈭� y}) k curl F = (\frac{鈭� (\cos (x) \sin (y))}{鈭� x} - \frac{鈭� (x^{2} e^{y})}{鈭� y}) k = (- sinx siny - x^{2} e^{y}) k Now, using curl form of green' s theorem to evaluate the integral : {鈭�}_{c}^{} F (x, y) . dx = 鈭� {鈭�}_{R} curl F . k dA = 鈭� {鈭�}_{R} (- sinx siny - x^{2} e^{y}) k . kdA = - 鈭� {鈭�}_{R} (sinx siny + x^{2} e^{y}) dA Therefore, the line integral as double integral is - 鈭� {鈭�}_{R} (sinx siny + x^{2} e^{y}) dA$

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91影视

Use the curl form of Green鈥檚 Theorem to write the line integral of F(x, y) about the unit circle as a double integral. Do not evaluate the integral.

Short Answer

Step by step solution

Step 1. Given information is:

Step 2. Line integral of f(x,y)

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