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Home Vector Calculus Independence of Path Examples Example 3 (Continued)
 
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Example 3 (Continued)

EXAMPLE 3 (Continued)

Find a potential function for the vector field

3x2 y2 i + (y2 + 2x3 y) j.

Step 1

Pick (0, 0) for the initial point.

Step 2

Let C be the rectangular curve from (0,0) to (0, y0) to (x0,y0), shown in Figure 13.3.2. We have already shown that

13_vector_calculus-132.gif

13_vector_calculus-131.gif

Figure 13.3.2

Step 3

A potential function is

13_vector_calculus-134.gif

As a check we may compute grad f.

13_vector_calculus-135.gif

 

We can get the same answer by choosing another curve in Step 2.

FIRST ALTERNATE SOLUTION

Step 2

Let C1 be the rectangular curve from (0, 0) to (x0, 0) to (x0,y0), shown in Figure 13.3.3.

13_vector_calculus-133.gif

Figure 13.3.3

Step 3

13_vector_calculus-136.gif

SECOND ALTERNATE SOLUTION

Step 2

Let C2 be the straight line from (0,0) to (x0,y0), shown in Figure 13.3.4. It has parametric equations

x = tx0, y = ty0, 0 ≤ r ≤ 1.

13_vector_calculus-138.gif

Figure 13.3.4

Step 3

13_vector_calculus-137.gif

Home Vector Calculus Independence of Path Examples Example 3 (Continued)

Last Update: 2006-11-25