Find the work done by 𝐅 over the curve in the direction of increasing t . 𝐅=x y 𝐢+y 𝐣-y z 𝐤

step 1 In this problem, we are given a force vector f and a position vector r of t for which we want to

Find the work done by 𝐅 over the curve in the direction of...

Find the work done by $\mathbf{F}$ over the curve in the
direction of increasing $t .$
$$
\begin{array}{l}{\mathbf{F}=x y \mathbf{i}+y \mathbf{j}-y z \mathbf{k}} \\ {\mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+t \mathbf{k}, \quad 0 \leq t \leq 1}\end{array}
$$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Line Integral of a Vector Field

The line integral calculates the work done by a vector field along a curve. It is defined as the integral of the dot product of the vector field and the differential displacement along the curve, which represents the accumulation of the force’s component in the direction of the path.

Parameterization of a Curve

Parameterizing a curve involves expressing the curve as a function of a single variable, typically denoted by t. This allows converting the line integral into a single-variable integral by substituting the parameterized functions into the vector field and computing the derivative of the position vector with respect to the parameter.

Dot Product in Line Integrals

The dot product in this context measures how much of the vector field is aligned with the direction of the curve at each point. By taking the dot product of the force field and the tangent vector to the curve, one obtains a scalar function that can be integrated over the interval of the parameter.

Evaluation of Definite Integrals

Once the line integral is reduced to a single-variable integral, the standard techniques of evaluating definite integrals, such as finding antiderivatives and applying the Fundamental Theorem of Calculus, are used to compute the total work done along the curve.

Transcript

Please give Ace some feedback