Find the potential function f for the field F. 17) F=(ey2−sinx)i+2xyey2j+k A) f(x,y,z)=cosx+xzey2+C B) f(x,y,z)=cosx+ey2+z+C C) f(x,y,z)=cosx+ey2(x+z)+C D) f(x,y,z)=cosx+xey2+z+C 3 18) F=z1i−3j−z2xk A) f(x,y,z)=z2x−3y+C B) f(x,y,z)=zx−3+C C) f(x,y,z)=zx−3y+C D) f(x,y,z)=zx+C

17.) The field F is given by The potential function f and the field F is related as F=gradf. Compare both sides,...

Resuelto Find the potential function f for the field F. 17)

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Pregunta: Find the potential function f for the field F. 17) F=(ey2−sinx)i+2xyey2j+k A) f(x,y,z)=cosx+xzey2+C B) f(x,y,z)=cosx+ey2+z+C C) f(x,y,z)=cosx+ey2(x+z)+C D) f(x,y,z)=cosx+xey2+z+C 3 18) F=z1i−3j−z2xk A) f(x,y,z)=z2x−3y+C B) f(x,y,z)=zx−3+C C) f(x,y,z)=zx−3y+C D) f(x,y,z)=zx+C
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Solución
Paso 1
17.) The field $F$ is given by
$\begin{aligned} F & = (e^{y^{2}} - \sin x) i + 2 x y e^{y^{2}} j + k \end{aligned}$

Explanation:
The potential function $f$ and the field $F$ is related as $F = \nabla f$ .

$\begin{matrix} \begin{aligned} \nabla f & = F \\ \frac{\partial f}{\partial x} i + \frac{\partial f}{\partial y} j + \frac{\partial f}{\partial z} k & = (e^{y^{2}} - \sin x) i + 2 x y e^{y^{2}} j + k \end{aligned} \end{matrix}$
Compare both sides,...
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Paso 2
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Respuesta
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Texto de la transcripción de la imagen:

Find the potential function

f

for the field

F

. 17)

F = (e y^{2} - sin x) i + 2 x ye y^{2} j + k

f (x, y, z) = cos x + x ze y^{2} + C

f (x, y, z) = cos x + e y^{2} + z + C

f (x, y, z) = cos x + e y^{2} (x + z) + C

f (x, y, z) = cos x + x e y^{2} + z + C

3 18)

F = \frac{1}{z} i - 3 j - \frac{x}{z ^{2}} k

f (x, y, z) = \frac{2 x}{z} - 3 y + C

f (x, y, z) = \frac{x}{z} - 3 + C

f (x, y, z) = \frac{x}{z} - 3 y + C

f (x, y, z) = \frac{x}{z} + C