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PAGE 1

Example: Conservative Vector Fields

\[ \vec{F} = \langle \underbrace{x^2 - ze^y}_{f}, \underbrace{y^3 - xze^y}_{g}, \underbrace{z^4 - xe^y}_{h} \rangle \]

Check for conservativeness by testing mixed partial derivatives:

  • is \( f_y = g_x \) ?
  • is \( g_z = h_y \) ?
  • is \( f_z = h_x \) ?
  • \( -ze^y = -ze^y \) yes
  • \( -xe^y = -xe^y \) yes
  • \( -e^y = -e^y \) yes
ALL 3 must be yes for \( \vec{F} = \nabla \phi \)

Finding the Potential Function

Now we know:

\[ \vec{F} = \langle f, g, h \rangle = \langle \phi_x, \phi_y, \phi_z \rangle \]
  1. 1 \( \phi_x = x^2 - ze^y \)
  2. 2 \( \phi_y = y^3 - xze^y \)
  3. 3 \( \phi_z = z^4 - xe^y \)
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From 1:

\[ \phi = \int (x^2 - ze^y) dx = \frac{1}{3}x^3 - xze^y + a(y, z) \quad \text{④} \]
Note: \( y, z \) are constants during integration with respect to \( x \). The term \( a(y, z) \) is a function that can depend on \( y, z \).

What is \( a \)?

Partial of 4 with \( y \) must be 2:

\[ \phi_y = -xze^y + \frac{\partial a}{\partial y} = \underbrace{y^3 - xze^y}_{ ext{②}} \rightarrow \frac{\partial a}{\partial y} = y^3 \quad \text{⑤} \]

Partial of 4 with \( z \) must be 3:

\[ \phi_z = -xe^y + \frac{\partial a}{\partial z} = \underbrace{z^4 - xe^y}_{ ext{③}} \rightarrow \frac{\partial a}{\partial z} = z^4 \quad \text{⑥} \]

Solving for \( a \)

Integrate 5 with \( y \):

\[ a = \frac{1}{4}y^4 + b(z) \]

Take partial with \( z \) and compare to 6:

\[ \frac{\partial a}{\partial z} = \frac{db}{dz} = z^4 \rightarrow b = \frac{1}{5}z^5 + C \]

So \( a = \frac{1}{4}y^4 + \frac{1}{5}z^5 + C \)