Q. 15 Use the same vector field as in ... [FREE SOLUTION]

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Chapter 14: Q. 15 (page 1132)

Use the same vector field as in Exercise 13, and compute the k-component of the curl of F(x, y).

Short Answer

Expert verified

$The k - component of the curl of F (x, y) is - sinx siny - x^{2} e^{y} .$

Step by step solution

Step 1. Given information is

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

Step 2. Curl of F

$The curl of a vector field F (x, y) = F_{1} (x, y) i + F_{2} (x, y) j in {鈩�}^{2} is defined as follows : curl F = (\frac{鈭� F_{2}}{鈭� x} - \frac{鈭� F_{1}}{鈭� y}) k Hence the k component of the curl of F (x, y) is, \frac{鈭� F_{2}}{鈭� x} - \frac{鈭� F_{1}}{鈭� y} For the vector field, F (x, y) = x^{2} e^{y} i + \cos (x) \sin (y) j F_{1} (x, y) = x^{2} e^{y} F_{2} (x, y) = \cos (x) \sin (y) Now, first find \frac{鈭� F_{2}}{鈭� x} and \frac{鈭� F_{1}}{鈭� y} Then, \frac{鈭� F_{2}}{鈭� x} = \frac{鈭� (\cos x \sin y)}{鈭� x} = - sinx siny and, \frac{鈭� F_{1}}{鈭� y} = \frac{鈭� (x^{2} e^{y})}{鈭� y} = x^{2} e^{y} Then, \frac{鈭� F_{2}}{鈭� x} - \frac{鈭� F_{1}}{鈭� y} = - sinx siny - x^{2} e^{y} Therefore, the k - component of the curl of F (x, y) is - sinx siny - x^{2} e^{y} .$

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91影视

Use the same vector field as in Exercise 13, and compute the k-component of the curl of F(x, y).

Short Answer

Step by step solution

Step 1. Given information is

Step 2. Curl of F

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