Calculate the work done by a particle moving under the influence of a force F =-(2z + y)â + x²yŷ + (z – x²)2 along a path defined byy = x² + 1, z = x² – 4ygoing from (0, 1, -4) to (1, 2, –7).

Answered: Calculate the work done by a particle…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Calculate the work done by a particle moving under the influence of a force F =
-(2z + y)â + x²yŷ + (z – x²)2 along a path defined by
y = x² + 1, z = x² – 4y
going from (0, 1, -4) to (1, 2, –7).

Expert Solution

Step 1

The general form of a vector-valued function in three dimensions is given by $F = F_{1} \hat{x} + F_{2} \hat{y} + F_{3} \hat{z}$ where $F_{1}, F_{2}$ and $F_{3}$ are the scalar functions of the variables x,y, and z. A vector-valued function is a rule that is used to fix or associate a vector to a particular point $(x, y, z)$ in space.

The line integral $\int_{C} \vec{F} \cdot d \vec{r}$ involves the integration over a specific path. The vector $d \vec{r} = d x \hat{x} + d y \hat{y} + d z \hat{z}$ represents the infinitesimal length over the path of integration. A line integral on a closed curve can be converted into a surface integral by Stokes theorem.