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Vector Calculus 3

An engineering math concept

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Jeremiah Idam
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30 views6 pages

Vector Calculus 3

An engineering math concept

Uploaded by

Jeremiah Idam
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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CURL OF A VCECTOR

• The curl of a vector field is a mathematical operation in vector calculus that describes the
rotation or circulation of the vector field at a given point. It is denoted by the symbol ∇× or curl.
The result of taking the curl of a vector field is another vector field.
• For a three-dimensional vector field F= (P,Q,R), the curl ∇×F is given by:
� � �
� × � = ����� = � � �
�� �� ��
�� �� ��

The physical interpretation of the curl is related to the rotation or circulation of a vector field. If
the curl at a point is zero, it indicates that the vector field is irrotational at that point. If the curl is
non-zero, it suggests that the vector field has rotation or circulation.
Example 1.
Consider a vector field H(x,y,z) = (yz,xz,xy), calculate its curl ∇×H.
Solution
Using

∴ curl ∇×H is given by:

� × � = (� − �)� + (� − �)� + (� − �)�


�� =0
So, the curl of the vector field H is zero. This implies that the vector field is irrotational,
meaning it does not exhibit rotation or circulation at any point in space.
Example:
Given
Determine the curl of F.
Solution
Using

=− 4�2 � − 2�2 � = (−4�2 , 0, − 2�2 )


TRY THIS
• Given show that
find the curl of a three-dimensional vector field given
G(x,y,z)= (y2,xz,−2yz). Comment on your result.

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