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 )

Note by Rishi Tiwari
3 weeks, 2 days ago

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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 - 3 weeks, 2 days ago

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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

Rishi Tiwari - 3 weeks, 2 days ago

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@Steven Chase Help me with this new problem about equilibria of pulleys

Rishi Tiwari - 1 week, 5 days ago

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@Steven Chase @Josh Silverman As shown in the following fig.in the given three arrangements A,B and C where pulleys are ideal,strings are light and inextensible,balls attached at the ends of strings are identical and surfaces where the balls rest are frictionless .Identify the nature of equilibria (i.e stable,unstable or neutral) of the arrangements . @Steven Chase @Josh Silverman Now in the figure shown we have to assume that pulley like in A arrangement is also in B and C arrangement but instead of inclined surfaces concavity in B and convexity in C is introduced as it was super hard for me to draw such figures with mouse .PLS give detailed solutions

Rishi Tiwari - 1 week, 5 days ago

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@Steven Chase and @Josh Silverman.Help me with this As shown in the following fig.in the given three arrangements A,B and C where pulleys are ideal,strings are light and inextensible,balls attached at the ends of strings are identical and surfaces where the balls rest are frictionless .Identify the nature of equilibria (i.e stable,unstable or neutral) of the arrangements . @Steven Chase @Josh Silverman Now in the figure shown we have to assume that pulley like in A arrangement is also in B and C arrangement but instead of inclined surfaces concavity in B and convexity in C is introduced as it was super hard for me to draw such figures with mouse .PLS give detailed solutions

Rishi Tiwari - 1 week, 5 days ago

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Thanks a lot can you recommend me something on books ,videos,websites,problems regarding the preparation of Indian National Physics Olympiad.

Rishi Tiwari - 3 weeks, 2 days ago

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I haven't been involved with those Olympiads, but there are many on this site who have. I'm sure they could give you some suggestions.

Steven Chase - 3 weeks, 1 day ago

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As shown in the following fig.in the given three arrangements A,B and C where pulleys are ideal,strings are light and inextensible,balls attached at the ends of strings are identical and surfaces where the balls rest are frictionless .Identify the nature of equilibria (i.e stable,unstable or neutral) of the arrangements . @Steven Chase @Josh Silverman Now in the figure shown we have to assume that pulley like in A arrangement is also in B and C arrangement but instead of inclined surfaces concavity in B and convexity in C is introduced as it was super hard for me to draw such figures with mouse .PLS give detailed solutions

Rishi Tiwari - 1 week, 5 days ago

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reply asap

Rishi Tiwari - 3 weeks, 2 days ago

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Help asap

Rishi Tiwari - 3 weeks, 2 days ago

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@Steven Chase

Rishi Tiwari - 3 weeks, 2 days ago

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@Josh Silverman @Steven Chase

Rishi Tiwari - 3 weeks, 2 days ago

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