Question

Consider the vector field\mathbf{F} \left( x, y, z \right) = \left( z+ y\right) \mathbf{i} + \left( 3 z+ x \right) \mathbf{j} + \left( 3 y+ x \right) \mathbf{k}.

a) Find a functionfsuch that\mathbf{F} = \\abla fandf(0,0,0) = 0.
f(x,y,z) =
b) Suppose C is any curve from\left( 0, 0, 0 \right)to\left( 1, 1, 1 \right).Use part a) to compute the line integral\int_{C} \mathbf{F} \cdot d\mathbf{r}

Answer 1

a) To find a function f such that
`\mathbf{F} = \\abla f and f(0,0,0) = 0 \\`, we need to find the potential function f(x, y, z) whose gradient is equal to
`\mathbf{F} \\`.

Given
`\mathbf{F} = (z + y)\mathbf{i} + (3z + x)\mathbf{j} + (3y + x)\mathbf{k} \\`, we can find the potential function f(x, y, z) by integrating the components of
`\mathbf{F}\\` with respect to their corresponding variables:

`\:f(x, y, z) = \int (z + y) \, dx + \int (3z + x) \, dy + \int (3y + x) \, dz \\`

Integrating each component:

`\:f(x, y, z) = (1)/(2)zx + yx + g_1(y, z) + (3)/(2)zy + g_2(x, z) + (3)/(2)yx + g_3(x, y) \\`

where g_1(y, z), g_2(x, z), and g_3(x, y) are arbitrary functions of their respective variables.

To satisfy the condition f(0, 0, 0) = 0, we set the arbitrary functions g_1(y, z), g_2(x, z), and g_3(x, y) to be zero:

`f(x, y, z) = (1)/(2)zx + yx + (3)/(2)zy + (3)/(2)yx \\`

Therefore, the function f(x, y, z) that satisfies
`\mathbf{F} = \\abla f \\` and f(0, 0, 0) = 0 is:

`f(x, y, z) = (1)/(2)zx + yx + (3)/(2)zy + (3)/(2)yx \\`

b) Now, we want to compute the line integral
`\int_C \mathbf{F} \cdot d\mathbf{r} \\` along any curve C from (0, 0, 0) to (1, 1, 1).

Using the fundamental theorem of line integrals, we can evaluate the line integral by evaluating the potential function f at the endpoints of the curve:

`\int_C \mathbf{F} \cdot d\mathbf{r} = f(1, 1, 1) - f(0, 0, 0) \\`

Substituting the values into the potential function:

`\int_C \mathbf{F} \cdot d\mathbf{r} = f(1, 1, 1) - f(0, 0, 0) \\ \\= \left((1)/(2) + 1 + (3)/(2) + (3)/(2)\right) - \left(0\right) = (11)/(2) \\`

Therefore, the line integral
`\int_C \mathbf{F} \cdot d\mathbf{r} \\` along any curve C from (0, 0, 0) to (1, 1, 1) is
`(11)/(2) \\`.