Solved: 1516 Confirm that is a potential function for F(r) on someregion, and state the

Chapter 15, Problem 16

(choose chapter or problem)

Confirm that $\phi$ is a potential function for F(r) on some region, and state the region.

(a) $\phi(x, y)=2 y^{2}+3 x^{2} y-x y^{3}$

F$(x, y)=\left(6 x y-y^{3}\right) i+\left(4 y+3 x^{2}-3 x y^{2}\right) j$

(b) $\phi(x, y, z)=x \sin z+y \sin x+\sin y$

F$x, y, z)=(\sin z+y \cos x) i+(\sin x+z \cos y) j$

$+(\sin y+x \cos z) k$

Equation Transcription:

𝜙

𝜙 $(\ x,\ y)\ \ =\ \ 2{y}^{2}\ \ +\ \ 3{x}^{2}y\ \ -\ \ x{y}^{3}$

$(x,\ y)\ \ =\ (6xy\ \ -\ \ {y}^{3})i\ \ +\ \ (4y\ \ +\ \ 3{x}^{2}\ \ -\ \ 3x{y}^{2})j$

𝜙 $(x,\ y,\ z)\ \ =\ \ x\ \ sin\ \ z\ \ +\ \ y\ \ sin\ \ x\ \ +\ \ \ \ sin\ \ y$

$(x,\ y,\ z)\ \ =\ (sin\ \ z\ \ +\ \ y\ \ cos\ \ x)i\ \ +\ \ (sin\ \ x\ \ +\ \ z\ \ cos\ \ y)j$

$+\ \ (sin\ \ y\ \ +\ \ x\ \ cos\ \ z)k$

Text Transcription:

phi

phi(x, y) = 2y^2 + 3 x^2y - xy^3

(x, y) = (6xy - y^3) i + (4y + 3 x^2 - 3xy^2) j

phi(x, y, z) = x sin z + y sin x + sin y

(x, y, z) = (sin z + y cos x) i + (sin x + z cos y) j

+(sin y + x cos z) k

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