Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.

Please verify your account before proceeding.

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 .

Donβt have an account? Sign Up