Determine whether or not the vector field is conservative. $\vec { F } ( x , y ) = 35 x ^ { 6 } y ^ { 6 } \hat { \mathbf { i } } + 30 x ^ { 7 } y ^ { 5 } \hat { \mathbf { j } }$
A) conservative
B) not conservative

Question

Quizplus · Accepted Answer

A vector field $\vec{F}(x, y) = M(x, y)\hat{\mathbf{i}} + N(x, y)\hat{\mathbf{j}}$ is conservative if and only if its curl is zero, which means $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$. For $\vec{F}(x, y) = 35x^6y^6\hat{\mathbf{i}} + 30x^7y^5\hat{\mathbf{j}}$, we find $\frac{\partial M}{\partial y} = 210x^6y^5$ and $\frac{\partial N}{\partial x} = 210x^6y^5$. Since these are equal, the vector field is conservative.

Determine Whether or Not the Vector Field Is Conservative F⃗(x,y)=35x6y6i^+30x7y5j^\vec { F } ( x , y ) = 35 x ^ { 6 } y ^ { 6 } \hat { \mathbf { i } } + 30 x ^ { 7 } y ^ { 5 } \hat { \mathbf { j } }F(x,y)=35x6y6i^+30x7y5j^​

Determine Whether or Not the Vector Field Is Conservative $\vec { F } ( x , y ) = 35 x ^ { 6 } y ^ { 6 } \hat { \mathbf { i } } + 30 x ^ { 7 } y ^ { 5 } \hat { \mathbf { j } }$