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Divergence and curl practice problems

∇ · F and ∇ × F, component by component. 21 problems with worked solutions; in 21 of them every equation is proved by a computer algebra system.

Find the divergence and curl of \( \displaystyle \mathbf{F} = \langle x^{2}, y^{2}, z^{2} \rangle \).

Problem 12.31easy✓ Every equation proved

Find the divergence and curl of \( \displaystyle \mathbf{F} = \langle y z, x z, x y \rangle \).

Problem 12.32easy✓ Every equation proved

Find the divergence and curl of \( \displaystyle \mathbf{F} = \langle \sin{\left(y \right)}, x \cos{\left(z \right)}, y \rangle \).

Problem 12.33easy✓ Every equation proved

Find the divergence and curl of \( \displaystyle \mathbf{F} = \langle x z^{2}, - y, x y z \rangle \).

Problem 12.34easy✓ Every equation proved

Find the divergence and curl of \( \displaystyle \mathbf{F} = \langle x^{2} y, - x y^{2}, z \rangle \).

Problem 12.35easy✓ Every equation proved

Find the divergence and curl of \( \displaystyle \mathbf{F} = \langle x y, y z, x z \rangle \).

Problem 12.36easy✓ Every equation proved

Find the divergence and curl of \( \displaystyle \mathbf{F} = \langle - 2 x y, 3 x^{2}, 3 x^{2} \rangle \).

Problem 12.67easy✓ Every equation proved

Find the divergence and curl of \( \displaystyle \mathbf{F} = \langle 2 y z, - 2 \sin{\left(y \right)}, 3 y^{2} \rangle \).

Problem 12.68easy✓ Every equation proved

Find the divergence and curl of \( \displaystyle \mathbf{F} = \langle 2 x, - z, 2 \cos{\left(z \right)} \rangle \).

Problem 12.69easy✓ Every equation proved

Find the divergence and curl of \( \displaystyle \mathbf{F} = \langle - 3 x, 3 \sin{\left(y \right)}, \sin{\left(y \right)} \rangle \).

Problem 12.70easy✓ Every equation proved

Find the divergence and curl of \( \displaystyle \mathbf{F} = \langle - 2 x, 2 x y z, x y z \rangle \).

Problem 12.71easy✓ Every equation proved

Find the divergence and curl of \( \displaystyle \mathbf{F} = \langle - 3 e^{x}, - 2 \cos{\left(z \right)}, 3 x y z \rangle \).

Problem 12.72easy✓ Every equation proved

Find the divergence and curl of \( \displaystyle \mathbf{F} = \langle - 2 e^{x}, 2 e^{x}, x y z \rangle \).

Problem 12.73easy✓ Every equation proved

Find the divergence and curl of \( \displaystyle \mathbf{F} = \langle - 2 x y, 3 y z, 2 y z \rangle \).

Problem 12.74easy✓ Every equation proved

Find the divergence and curl of \( \displaystyle \mathbf{F} = \langle - x, - x z, 2 y^{2} \rangle \).

Problem 12.75easy✓ Every equation proved

Find the divergence and curl of \( \displaystyle \mathbf{F} = \langle 2 x y, - 3 e^{x}, 3 z \rangle \).

Problem 12.76easy✓ Every equation proved

Find the divergence and curl of \( \displaystyle \mathbf{F} = \langle - 2 x y, 3 x y, 3 x z \rangle \).

Problem 12.77easy✓ Every equation proved

Find the divergence and curl of \( \displaystyle \mathbf{F} = \langle x z, - z^{2}, 3 e^{x} \rangle \).

Problem 12.78easy✓ Every equation proved

Find the divergence and curl of \( \displaystyle \mathbf{F} = \langle - 2 \sin{\left(y \right)}, - y z, z \rangle \).

Problem 12.79easy✓ Every equation proved

Find the divergence and curl of \( \displaystyle \mathbf{F} = \langle - 2 \sin{\left(y \right)}, 2 x y z, 2 \cos{\left(z \right)} \rangle \).

Problem 12.80easy✓ Every equation proved

Find the divergence and curl of \( \displaystyle \mathbf{F} = \langle 2 x z, 2 y, y z \rangle \).

Problem 12.81easy✓ Every equation proved