The image below shows the vector field with the magnitude of the curl drawn as a surface above it: The green arrow is the curl at $$(\pi/4, \pi/4)$$. Find out what you can do.

It is indicating how much the field is circulating the given area ‘ds‘. It's difficult to plot, because the vector field blows up at the origin. $\mathbf{F}(x, y, z) = P(x, y, z) \vec{i} + Q(x, y, z) \vec{j} + R(x, y, z) \vec{k}$, $\mathbf{F} (x, y, z) = P(x, y, z) \vec{i} + Q(x, y, z) \vec{j} + R(x, y, z) \vec{k}$, $\mathrm{curl} (\mathbf{F}) = \nabla \times \mathbf{F} = \left ( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right ) \vec{i} + \left ( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right ) \vec{j} + \left ( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right ) \vec{k}$, $\mathrm{curl} (\mathbf{F}) = \nabla \times \mathbf{F}$, $\nabla = \frac{\partial}{\partial x} \vec{i} + \frac{\partial}{\partial y} \vec{j} + \frac{\partial}{\partial z} \vec{k}$, $\mathbf{F} = P \vec{i} + Q \vec{j} + R \vec{k}$, $\mathbf{F} (x, y, z) = f(x) \vec{i} + g(y) \vec{j} + h(z) \vec{k}$, $\mathbf{F}(x, y, z) = x^2y \vec{i} + xy^2z \vec{j} + xe^z \vec{k}$, $\mathbf{F}(x, y, z) = \sin (xz) \vec{i} + \cos(xyz) \vec{j} + xyz \vec{k}$, $\frac{\partial Q}{\partial z} -xy \sin (xyz)$, $\frac{\partial P}{\partial z} = x \cos (xz)$, $\frac{\partial Q}{\partial x} = -yz \sin (xyz)$, Creative Commons Attribution-ShareAlike 3.0 License.

Find the curl of the vector field $\mathbf{F}(x, y, z) = x^2y \vec{i} + xy^2z \vec{j} + xe^z \vec{k}$. Let $f(x,y,z)$ define a function with continuous first and second partial derivatives. If you're seeing this message, it means we're having trouble loading external resources on our website. The curl of the vector field E is represented as ∇ × E. And finally, the representation of the curl of the vector field is given as-, Suggested Community: Electromagnetics for GATE & ESE, © The Right Gate | Conceptualized by SNAXZER. David Smith (Dave) has a B.S.

dr, where F = xy, yz, zx, and C is the triangle with vertices (1, 0, 0), (0, 1, 0), and (0, 0, 1), oriented clockwise when viewed from above. \begin{align} \quad \mathrm{div} (\mathbf{F}) = \nabla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \end{align}, \begin{align} \quad \mathrm{curl} (\mathbf{F}) = \nabla \times \mathbf{F} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z}\\ P & Q & R \end{vmatrix} = \left ( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right ) \vec{i} + \left ( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right ) \vec{j} + \left ( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right ) \vec{k} \end{align}, \begin{align} \quad \quad \mathrm{curl} (\mathbf{F}) = \nabla \times \mathbf{F} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ f(x) & g(y) & h(z) \end{vmatrix} = \left ( \frac{\partial}{\partial y} h(z) - \frac{\partial}{\partial z} g(y) \right ) \vec{i}+ \left ( \frac{\partial}{\partial z} f(x) - \frac{\partial}{\partial x} h(z) \right ) \vec{k} + \left ( \frac{\partial}{\partial x} g(y) - \frac{\partial}{\partial y} f(x) \right ) \vec{j} = 0\vec{i} + 0 \vec{j} + 0 \vec{k} \end{align}, \begin{align} \quad \mathrm{curl} (\mathbf{F}) = (0 - xy^2) \vec{i} + (0 - e^z) \vec{j} + (y^2 - x^2) \vec{k} \\ \quad \mathrm{curl} (\mathbf{F}) = -xy^2 \vec{i} - e^z \vec{j} + (y^2z - x^2) \vec{k} \end{align}, \begin{align} \quad \mathrm{curl} (\mathbf{F}) = (xz + xy \sin (xyz)) \vec{i} + (x\cos(xz) - yz) \vec{j} + (-yz \sin (xyz)) \vec{k} \\ \end{align}, Unless otherwise stated, the content of this page is licensed under. Let ${F}=\langle x^2y,y z^2,z y^2\rangle.$ Either find a vector field ${G}$ such that ${F}=\mathop{curl} {G}$ or show that no such ${G}$ exists. Not sure what college you want to attend yet? , is a vector whose magnitude is the maximum net circulation of A per unit area as the area tends to zero and whose direction is the normal direction of the area when the area is oriented to make the net circulation maximum!.