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12.03.2022

If vector A=xz³î-2x²yj+2yz⁴k find CURL A at the point [1, -1, 1]

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09.07.2023, solved by verified expert

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Given that

its curl is

so that at the point (1, -1, 1), the curl is

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Mathematics

If vector A=xz³î-2x²yj+2yz⁴k find CURL A at the point [1, -1, 1]

Step-by-step answer

P Answered by Specialist

1

Given that

$\vec A = xz^3 \, \vec\imath - 2x^2 y \, \vec\jmath + 2yz^4 \, \vec k$

its curl is

$\displaystyle \nabla\times\vec A = \left(\frac{\partial\left(2yz^4\right)}{\partial y} - \frac{\partial\left(-2x^2y\right)}{\partial z}\right) \, \vec\imath - \left(\frac{\partial\left(2yz^4\right)}{\partial x} - \frac{\partial\left(xz^3\right)}{\partial z}\right) \, \vec\jmath \\ ~~~~~~~~~~~~ + \left(\frac{\partial\left(-2x^2y\right)}{\partial x} - \frac{\partial\left(xz^3\right)}{\partial y}\right) \, \vec k$

$\nabla\times\vec A = 2z^4 \, \vec\imath + 3xz^2 \, \vec\jmath - 4xy \, \vec k$

so that at the point (1, -1, 1), the curl is

$\nabla\times\vec A \bigg|_{(x,y,z)=(1,-1,1)} = \boxed{2 \, \vec\imath + 3 \, \vec\jmath + 4 \, \vec k}$

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English

Dryope by Thomas Bulfinch

Dryope (DRY-oh-pee) was the wife of Andraemon (an-DREE-mun), beloved by her husband and happy about the birth of her first child. One day she and her sister, lole (EYE-oh-lee), strolled to the bank of a stream that sloped gradually down to the water's edge. Dryope carried her child, a precious burden, at her bosom and nursed him as she walked through the tall grass that grew along the river.

Near the water grew a lotus tree, full of purple flowers. Dryope gathered some and offered them to the baby, and lole was about to do the same, when she noticed blood dropping from the places where her sister had broken them off at the stem. The plant was none other than Lotus, who had been changed into this form by the gods in order to save the goddess from a cruel pursuer. This they learned from the country people when it was too late.

Dryope, horror-stricken when she perceived what she had done, would gladly have run from the spot but found her feet rooted to the ground. She tried to pull them away but could move nothing but her upper limbs. The woodiness crept upward and was slowly invading her body. In anguish she attempted to tear her hair but found her hands filled with leaves. The infant felt his mother's bosom begin to harden and the milk cease to flow. lole looked on at the sad fate of her sister and could do nothing to help. She embraced the growing trunk, as if she would hold back the advancing wood, and would gladly have been enclosed in the same bark.

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Adapted from Bulfinch's version of the ancient Greek myth, "Dryope," 1855.

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