Question

Consider the vector field F(x,y,z)=xi+yj+zk.

a) Find a function f such that F=∇f and f(0,0,0)=0.

f(x,y,z)=

b) Use part a) to compute the work done by F on a particle moving along the curve C given by r(t)=(1+4sin⁡t)i+(1+4sin2⁡t)j+(1+sin3⁡t)k,0≤t≤π2.

Work =

Answer 1

In order to find a function f such that F = ∇f, take the partial derivatives of f and equate them to the components of F. The work done by F on a particle moving along the curve C can be computed by evaluating the line integral ∫F⋅dr over the curve C.

Step-by-step explanation:

In order to determine a function f such that F = ∇f, where F is a vector field, we can hypothesize a function f(x, y, z) = ax^2 + by^2 + cz^2, introducing constants a, b, and c.

Deriving the partial derivatives of f, we acquire ∂f/∂x = 2ax, ∂f/∂y = 2by, and ∂f/∂z = 2cz.

Setting these derivatives equal to the corresponding components of F, we get 2ax = x, 2by = y, and 2cz = z.

Solving for a, b, and c reveals a = 1/2, b = 1/2, and c = 1/2. Thus, f(x, y, z) = 1/2(x^2 + y^2 + z^2).

To compute the work done by F on a particle following curve C, we evaluate the line integral ∫F⋅dr over C.

Substituting C's parametric equations into F, we get F(t) = (1 + 4sin⁡t)i + (1 + 4sin^2⁡t)j + (1 + sin^3⁡t)k, where 0 ≤ t ≤ π/2. By calculating the dot product F⋅dr and integrating over C, we obtain the work.

Answer 2

To find the function f(x, y, z) such that F = ∇f, integrate each component of F separately and find the constant of integration using the given conditions. Then, to compute the work done by F on a particle moving along the curve C, evaluate the line integral of F dot dr along C.

Step-by-step explanation:

To find a function f such that F = ∇f, we can integrate each component of F separately with respect to its respective variable. So, f(x, y, z) = ∫xi dx + ∫yj dy + ∫zk dz. Integrating each component gives f(x, y, z) = 1/2x^2 + 1/2y^2 + 1/2z^2 + C, where C is the constant of integration. Given that f(0, 0, 0) = 0, we can substitute the values into f to find C: 0 = 1/2(0)^2 + 1/2(0)^2 + 1/2(0)^2 + C. Solving for C, we get C = 0. Therefore, f(x, y, z) = 1/2x^2 + 1/2y^2 + 1/2z^2.

To compute the work done by F on the particle moving along the curve C, we need to evaluate the line integral of F dot dr along C. Let's denote the position vector r(t) = (1 + 4sin(t))i + (1 + 4sin^2(t))j + (1 + sin^3(t))k. We can find the velocity vector dr/dt by taking the derivative of r with respect to t. Then, compute F dot dr and integrate it with respect to t from 0 to π/2 to find the work done.