Chapter 1. calc_tutorial_16_3_12

1.1 Problem Statement

{4,6,8,10,12,14,16,18,20,22}
$a/2

Find a potential function for F or determine that F is not conservative.

F = <cos z, $ay, −x sin z>

1.2 Step 1

Question Sequence

Question 1.1

Recall the cross-partial property of conservative vector fields.

F = <F1, F2, F3> is conservative if and only if (∂F1 / ∂y) = (∂F2 / ∂x), (∂F2 / ∂z) = (∂F3 / ∂y), and (∂F3 / ∂x) = (∂F1 / ∂z).

Use this property to check if F = <cos z, $ay, −x sin z> is conservative.

(∂F1 / ∂y) = 1Wh3cvJ2xF4= (∂F2 / ∂x) = 1Wh3cvJ2xF4=

(∂F2 / ∂z) = 1Wh3cvJ2xF4= (∂F3 / ∂y) = 1Wh3cvJ2xF4=

(∂F3 / ∂x) = uwmOfx3Ox03FF3+CIQuASg== (∂F1 / ∂z) = uwmOfx3Ox03FF3+CIQuASg==

Thus F = <cos z, $ay, −x sin z> bfdtn5PFpMarVC8x2zfiEQ== conservative.

Correct.
Incorrect.

1.3 Step 2

Question Sequence

Question 1.2

We wish to find a potential function, V(x, y, z), such that F = ∇V. Thus cos z = F1 = (∂V / ∂x).

Integrate (∂V / ∂x) = cos z with respect to x, holding y and z fixed.

= dCanhwbD+4KNCW01s2147L1bTHryaQpKO0Hw0A== + C(y, z)

We need to determine C(y, z). To do so, differentiate this expression with respect to y.

(∂V / ∂x) = 1Wh3cvJ2xF4= + Cy(y, z)

Correct.
Incorrect.

1.4 Step 3

Question Sequence

Question 1.3

Since F = ∇V, we have $az = F2 = (∂V / ∂y) = Cy(y, z).

Integrate Cy(y, z) = $ay with respect to y, holding z fixed.

= iSba6t70dtA=y2 + D(z)

We now have V(x, y, z) = x cos z + iSba6t70dtA=y2 + D(z), and need to determine D(z). To do so, differentiate this expression with respect to z.

(∂V / ∂z) = iSW7SrEvKkixlfQq8AcQQ/iif9TxwFRtsxbSPAn3H5u/Pn+yXLrelg== + Dz(z)

Correct.
Incorrect.

1.5 Step 4

Question Sequence

Question 1.4

Since −x sin z = F3, use the condition (∂V / ∂z) = F3 to solve for Dz(z).

Dz(z) = 1Wh3cvJ2xF4=

Integrate Dz(z) with respect to z.

D(z) =

s45ImIzFoDCqRKqWGPmRk+Ksezm9UwEaU6ur1edSyV/Fw1szQZLOZqRwyWMoxSwURWoY8SHbQWacwEHVNflRU6vPhyYp6dHZOmagHf5aHl7XQiD+GDQAaX93KogvCcRd6VuJ4oFB4wd4O8MnyOC070icm+2GsVSljzEw5VM+opxigA/W9Ty8/oilmYKRh25s1J/fjf9pAtZM0dBkzJFsueVNmY0/zY+N2S0dY+bnVlRVF9DsYqvDt0GrrUTL3szwfQUWnXllItjyTk9JJTJzh9C2E+NzC39Rve/1BxpdV+sJn5ioJlPuXQvdSYagIU38fXr4GrXAzmP3SWf7tnD5EG3G6526QVuc1Bbl0byxbWMqu5CqtqfZXhCwVUIPDIvNJ4Py9eKjioCkc8RNEM3LzUq01QETz5TR1FAezQsy9hWmeTjQ1HCozPbEdJfenyf6GnuwW3B5OKxuzfwRN2nMXLrUALQ+77OLkDqBynw631TTuCNICm10NiRhljaZ2OtLlyXpxxVyW60e48kVBRFgUX+5FIPVLQ3Ok9e9kOthoujeF+OTPmTNCFuF5h+sp43ACFH+eSxoiJnRi610XJVeLElRiV9ECiT8MhI4meo+fmi0TRbuMLZS+yimWl6R/UlGLS9PDrqSgDwGirsmJY6S5uK2JOjc2zQUlC9mvZtS8rE4AYpj1cDzfP83Cx6TnbfbqQP24kULhlgO1ADNnTwKSMqlRQu/X8Pyb61NM6UQdJ6AfdwDV+7R1pXSx4NXkAmsw3GNICF6lP8TdFlL+Rkz2tGjg7slsgw9a8w4CNOIyMpnmkJKFetGsdzVDKef3Dp7n/aWcwxzKjxMTaBZSLE3DG/6XntVmfhbaXBdQfO2J5wI4GVgsqsa/jtNW7X8OvfhAQhd8M1A/gVvEY8a5Wb5OClNNR2hgE5/CApWzYPdzVBPGZAP4ZHomVR9dIwspR/OPQaztHTc23yx8Fus

= 1Wh3cvJ2xF4= + C

Thus the general potential function is V(x, y, z) = ugizl6JFBzxOeZzXWxrB0V5aobip69B5 + $by2 + C.

Correct.
Incorrect.