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In this lesson, we will learn how to examine gradient vector fields produced from a multivariate function.
Q1:
Find the gradient of π ( π₯ , π¦ ) = π₯ + π¦ β 1 ο¨ ο¨ .
Q2:
Compute the gradient for π ( π₯ , π¦ , π§ ) = π₯ π¦ π§ . s i n
Q3:
Compute the gradient for π ( π₯ , π¦ , π§ ) = β π₯ + π¦ + π§ . 2 2 2
Q4:
Find the gradient of π ( π₯ , π¦ ) = 1 π₯ + π¦ ο¨ ο¨ .
Q5:
Compute the gradient for π ( π₯ , π¦ , π§ ) = π₯ π . ο¨ ο ο
Q6:
Suppose π€ = πΉ ( π ( π₯ , π¦ ) ) with π = ( π₯ + π¦ , π₯ β π¦ , π₯ π¦ ) ο¨ ο¨ ο¨ ο¨ . Express the gradient β π€ οΌ π , β 2 3 ο (viewed as a 1 Γ 2 matrix) in terms of the 1 Γ 3 matrix β πΉ ( π ) , where π = π οΌ π , β 2 3 ο , and a matrix of partial derivatives of π .
Q7:
For π ( π₯ , π¦ , π§ ) = π§ π₯ + π¦ 2 2 in Cartesian coordinates, find β π in cylindrical coordinates.
Q8:
Find the gradient of π ( π₯ , π¦ ) = π₯ π¦ . l n
Q9:
Compute the gradient for π ( π₯ , π¦ , π§ ) = π₯ + π¦ + π§ . ο¨ ο¨ ο¨
Q10:
Suppose π€ = πΉ ( π ( π₯ , π¦ ) ) with π = ( π , π , π ) 1 2 3 and that π = π ( π ) for a point π β β 2 . Express the gradient β π€ ( π ) (viewed as a 1 Γ 2 matrix) in terms of the 1 Γ 3 matrix β πΉ ( π ) and a matrix of partial derivatives of π .
Q11:
Compute the gradient of the function π ( π₯ , π¦ ) = π₯ π 2 π¦ .
Q12:
Find a function so that the vector field is a gradient field.
Q13:
Compute the gradient of the function π ( π₯ , π¦ ) = 2 π₯ + 5 π¦ .
Q14:
We explore an example where a vector field F = β¨ πΉ , πΉ β© 1 2 satisfies π πΉ π π¦ β π πΉ π π₯ = 0 1 2 but does not come from a potential. On the plane with the origin removed, consider the vector field F ( π₯ , π¦ ) = ο β π¦ π₯ + π¦ , π₯ π₯ + π¦ ο 2 2 2 2 .
On the (open) half-plane π₯ > 0 , we can define the angle function π ( π₯ , π¦ ) = ο» π¦ π₯ ο a r c t a n . This is well defined and gives a value between β π 2 and π 2 . What is the gradient β π ?
Using the figure shown, use π above to define the angle function π ( π₯ , π¦ ) 1 on the region π¦ > 0 by a suitable composition with a π 2 rotation.
What is β π ( π₯ , π¦ ) 1 ?
Since π and π 1 agree on the quadrant π₯ > 0 , π¦ > 0 , we can define the angle π ( π₯ , π¦ ) at each point of the union with values between β π 2 and 3 π 2 . Using this, what is οΈ β πΆ F r d , where πΆ is the arc of the unit circle from οΏ 1 2 , β β 3 2 ο to οΏ β 1 β 2 , 1 β 2 ο ? Answer in terms of π .
In the same way, we can define π 2 on the half-plane π₯ < 0 and π 3 on π¦ < 0 . Hence, evaluate the line integral οΈ β πΆ F r d around the circle of radius β 2 , starting from π ( 1 , β 1 ) and going counterclockwise back to π , stating your answer in terms of π .
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