Conservative fields and potential functions practice problems
Test whether a field is conservative, find a potential function, and evaluate a line integral by the fundamental theorem. 10 problems with worked solutions; in 10 of them every equation is proved by a computer algebra system.
Is \( \displaystyle \mathbf F = \left(y + z\right)\mathbf i + \left(2 x + z\right)\mathbf j + \left(x + y\right)\mathbf k \) conservative? If so, find a potential function \( \displaystyle f \).
Is \( \displaystyle \mathbf F = \left(y e^{x y}\right)\mathbf i + \left(x e^{x y} + x\right)\mathbf j \) conservative? If so, find a potential function \( \displaystyle f \).
Is \( \displaystyle \mathbf F = \left(3 x^{2} + y^{2}\right)\mathbf i + \left(2 x y + x\right)\mathbf j \) conservative? If so, find a potential function \( \displaystyle f \).
Show that \( \displaystyle \mathbf F = \left(y z\right)\mathbf i + \left(x z\right)\mathbf j + \left(x y\right)\mathbf k \) is conservative, find a potential function \( \displaystyle f \), and evaluate \( \displaystyle \int_C \mathbf F \cdot d\mathbf r \) along any path from \( \displaystyle (1, 2, 2) \) to \( \displaystyle (0, 1, 2) \).
Show that \( \displaystyle \mathbf F = \left(2 x + z\right)\mathbf i + \left(2 y\right)\mathbf j + \left(x + 2 z\right)\mathbf k \) is conservative, find a potential function \( \displaystyle f \), and evaluate \( \displaystyle \int_C \mathbf F \cdot d\mathbf r \) along any path from \( \displaystyle (1, 0, 0) \) to \( \displaystyle (2, 2, 0) \).
Show that \( \displaystyle \mathbf F = \left(y z e^{x}\right)\mathbf i + \left(z e^{x}\right)\mathbf j + \left(y e^{x}\right)\mathbf k \) is conservative, find a potential function \( \displaystyle f \), and evaluate \( \displaystyle \int_C \mathbf F \cdot d\mathbf r \) along any path from \( \displaystyle (2, 2, 1) \) to \( \displaystyle (2, 2, -1) \).
Show that \( \displaystyle \mathbf F = \left(y + z\right)\mathbf i + \left(x + z\right)\mathbf j + \left(x + y\right)\mathbf k \) is conservative, find a potential function \( \displaystyle f \), and evaluate \( \displaystyle \int_C \mathbf F \cdot d\mathbf r \) along any path from \( \displaystyle (2, 1, 2) \) to \( \displaystyle (2, 1, -1) \).
Show that \( \displaystyle \mathbf F = \left(y e^{x y}\right)\mathbf i + \left(x e^{x y}\right)\mathbf j \) is conservative, find a potential function \( \displaystyle f \), and evaluate \( \displaystyle \int_C \mathbf F \cdot d\mathbf r \) along any path from \( \displaystyle (0, 2) \) to \( \displaystyle (1, 2) \).
Show that \( \displaystyle \mathbf F = \left(y z\right)\mathbf i + \left(x z\right)\mathbf j + \left(x y\right)\mathbf k \) is conservative, find a potential function \( \displaystyle f \), and evaluate \( \displaystyle \int_C \mathbf F \cdot d\mathbf r \) along any path from \( \displaystyle (1, 0, 2) \) to \( \displaystyle (1, 0, -1) \).
Show that \( \displaystyle \mathbf F = \left(y e^{x}\right)\mathbf i + \left(e^{x} + \cos{\left(y \right)}\right)\mathbf j \) is conservative, find a potential function \( \displaystyle f \), and evaluate \( \displaystyle \int_C \mathbf F \cdot d\mathbf r \) along any path from \( \displaystyle (2, -1) \) to \( \displaystyle (2, 0) \).