Partial Derivative: Finding the vector on a scalar field at point (3,5)

[fluxer5]

Homework Statement

A scalar field is given by the function: ∅ = 3x²y + 4y²
a) Find del ∅ at the point (3,5)
b) Find the component of del ∅ that makes a -60^o angle with the axis at the point (3,5)

Homework Equations

del ∅ = d∅/dx + d∅/dy

The Attempt at a Solution

I completed part a:
del ∅ = (6xy+4y²)$\hat{i}$ + (3x²+8y)$\hat{j}$ = 120$\hat{i}$ + 67$\hat{j}$ (this answer is correct)

I am having trouble with part b. My gut feeling is that I need to take a dot product; project the vector from 'part a' onto the vector that makes "a -60^o angle with the axis."

Assuming the equation is |a||b|cosθ,

• I think a = 120$\hat{i}$ + 67$\hat{j}$
• I'm not sure what b is. Maybe a unit vector?
• I'm thinking θ is 60^o + atan(67/120)

Also, I'm assuming the "axis" that the problem refers to is the x-axis. The answer is -13.02.

Thank you for any help.

EDIT: Oops. 120i + 67j is not correct; it's 90i + 67j. |90i + 67j|cos(60+atan(67/90)) = -13.02

 

[SammyS]

Homework Statement

A scalar field is given by the function: ∅ = 3x²y + 4y²
a) Find del ∅ at the point (3,5)
b) Find the component of del ∅ that makes a -60^o angle with the axis at the point (3,5)

Homework Equations

del ∅ = d∅/dx + d∅/dy

The Attempt at a Solution

I completed part a:
del ∅ = (6xy+4y²)$\hat{i}$ + (3x²+8y)$\hat{j}$ = 120$\hat{i}$ + 67$\hat{j}$ (this answer is correct)

I am having trouble with part b. My gut feeling is that I need to take a dot product; project the vector from 'part a' onto the vector that makes "a -60^o angle with the axis."

Assuming the equation is |a||b|cosθ,

• I think a = 120$\hat{i}$ + 67$\hat{j}$
• I'm not sure what b is. Maybe a unit vector?
• I'm thinking θ is 60^o + atan(67/120)

Also, I'm assuming the "axis" that the problem refers to is the x-axis. The answer is -13.02.

Thank you for any help.

EDIT: Oops. 120i + 67j is not correct; it's 90i + 67j. |90i + 67j|cos(60+atan(67/90)) = -13.02

Hello luxer5. Welcome to PF !

$\cos(\theta)\hat{i}+\sin(\theta)\hat{j}$ is a vector that makes an angle of θ with the positive x-axis. It also happens to have a magnitude of 1 .