
#1
Apr412, 02:27 AM

P: 20

1. The problem statement, all variables and given/known data
Is F = (2ye^x)i + x(sin2y)j + 18k a gradient vector field? 3. The attempt at a solution Yeah I just don't know...I started to find some partial derivatives but I really don't know what to do here. Please help! 



#2
Apr412, 04:50 AM

Sci Advisor
HW Helper
Thanks
P: 26,167

hi calculusisrad!
(try using the X^{2} button just above the Reply box ) learn: div(curl) = 0, curl(grad) = 0 does that help? 



#3
Apr412, 05:57 AM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 38,899

For any g(x,y), [itex]\nabla g= \partial g/\partial x\vec{i}+ \partial g/\partial y\vec{j}+ \partial g/\partial z\vec{k}[/itex].
So is there a function g such that [tex]\frac{\partial g}{\partial x}= 2ye^x[/tex] [tex]\frac{\partial g}{\partial z}= 18[/tex] and [tex]\frac{\partial g}{\partial y}= x sin(2y)[/tex]? One way to answer that is to try to find g by finding antiderivatives. Another is to use the fact that as long as the derivatives are continuous (which is the case here), the mixed second derivatives are equal: [tex]\frac{\partial}{\partial x}\left(\frac{\partial g}{\partial y}\right)= \frac{\partial}{\partial y}\left(\frac{\partial g}{\partial x}\right)[/tex] Is [tex]\frac{\partial x sin(2y)}{\partial x}= \frac{\partial 2ye^x}{\partial y}[/tex]? etc. 



#4
Apr412, 10:33 PM

P: 20

Vector calc, gradient vector fields
Thank you soo much :)
I am still confused, though. So if I can prove that dg/dxy = dg/dyx and dg/dxz = dg/dzx and dg/dyz = dg/dzy , I will have proved that F is a gradient vector field, correct??? I found that dg/dxy = sin2y, and d/dyx = 2e^x. Since they are not equal, that means that F is not a gradient vector field? Thanks 



#5
Apr512, 04:52 AM

Sci Advisor
HW Helper
Thanks
P: 26,167

hi calculusisrad!
(have a curly d: ∂ and try using the X^{2} and X_{2} buttons just above the Reply box ) this is because you're actually proving that curlF = 0, and if F is a gradient, then F = ∇φ, and so curlF = (curl∇)φ = 0 


Register to reply 
Related Discussions  
How to evaluate gradient of a vector? or del operator times a vector  General Physics  1  
Hard time visualizing gradient vector vs. tangent vector.  Calculus  3  
Can all vector fields be described as the vector Laplacian of another vector field?  Calculus  5  
Vector Calc Homework Help! Divergence Free Vector Fields  Calculus & Beyond Homework  0  
calc III vector fields  Calculus & Beyond Homework  2 