Find the work done by 𝐅 over the curve in the direction of increasing t. 𝐅=6 z 𝐢+y^2 j+12 x

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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 $t$.

$$
\begin{aligned}
& \mathbf{F}=6 z \mathbf{i}+y^2 \mathrm{j}+12 x \mathbf{k} \\
& \mathbf{r}(t)=(\sin t) \mathrm{i}+(\cos t) \mathrm{j}+(t / 6) \mathbf{k}, \quad 0 \leq t \leq 2 \pi
\end{aligned}
$$

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Key Concepts

Line Integral in a Vector Field

The line integral of a vector field is a fundamental concept in vector calculus that computes the work done by a field when moving a particle along a curve. It involves summing the contribution of the field's force component along the direction of the path over the entire curve, typically using the dot product between the field and the differential displacement vector.

Parametrization of a Curve

Parametrizing a curve involves expressing the coordinates of points on the curve as functions of a parameter. This approach simplifies the process of integration by transforming the line integral into a single-variable integral, where the parameter represents the position along the curve and its interval defines the start and end of the path.

Differential Displacement and Dot Product

The differential displacement vector represents an infinitesimal segment of the curve, and taking its dot product with the vector field quantifies the field's component in the direction of motion. Integrating this dot product along the curve provides the cumulative work done as the particle moves through the field.

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