Vector analysis

Draw a circle in the xy plane .At a few representative points draw the vector v tangent to the circle ,pointing in the clockwise direction.By comparing adjacent vectors,determine the sign of dv sub x/dy and dv sub y/dx .According to cartesian construct of curl that is cross product expansion.What is the direction curl of vector v?Explain how it illustrates the geometrical interpretation of curl.(Here d/dx and d/dy stands for partial derivatives and sub x and y means the x and y component of vector v )

#Calculus

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Comments

Define some terms on the unit circle (assuming the tangent vector $\vec{v}$ is clockwise).

$\large{x = cos \theta \\ y = sin \theta \\ z = 0 \\ v_x = sin \theta = y \\ v_y = -cos \theta = -x \\ v_z = 0}$

Take partial derivatives:

$\large{\frac{\partial{v_x}}{\partial{y}} = 1 \\ \frac{\partial{v_y}}{\partial{x}} = -1}$

Suppose the curl of $\vec{v}$ is $\vec{C}$ :

$\large{C_x = \frac{\partial{v_z}}{\partial{y}} - \frac{\partial{v_y}}{\partial{z}} = 0 \\ C_y = -\frac{\partial{v_z}}{\partial{x}} + \frac{\partial{v_x}}{\partial{z}} = 0 \\ C_z = \frac{\partial{v_y}}{\partial{x}} - \frac{\partial{v_x}}{\partial{y}} = -2}$

The curl of $\vec{v}$ is therefore a vector in the $- z$ direction, which agrees with the well-known right-hand rule.

Steven Chase - 2 years, 9 months ago

Thanks a lot you always help me I did the same way by introducing same parameters in the circle it assures that my attempt was right.As author has directly written the vector v even making v sub z non zero but that makes sense because choice of v sub z should be in a manner that C sub x and C sub y should be zero that is derivatives should cancel out to zero in order to make consistency with right hand rule