Vector Calculus, Unit normal to surface help

In summary, we are asked to find the gradient of the function \phi(x,y,z)=x^3+2xy+yz^3 at the point P=(1,1,2) and the direction of the unit normal to the surface \phi(x,y,z)=11 at P. After working out the gradient, we got 5i+10j+12k and calculated the magnitude to be √269. This means the unit normal to the surface is \frac{5i+10j+12k}{√269}. The value of 11 does not affect the normal to the surface, only the exact position.
  • #1
tarwe
2
0

Homework Statement


If [itex]\phi[/itex](x,y,z) = x3 + 2xy +yz3 find [itex]\nabla[/itex][itex]\phi[/itex] at the point P=(1,1,2) and direction of the unit normal to the surface [itex]\phi[/itex](x,y,z) = 11 at P.

Homework Equations





The Attempt at a Solution


Worked out [itex]\nabla[/itex][itex]\phi[/itex] to be 5i + 10j + 12k
Got |[itex]\nabla[/itex][itex]\phi[/itex]|= √256

so the unit normal to surface is surely [itex]\frac{5+10+12}{√269}[/itex]
but how does the =11 bit make a difference?

Thanksss
 
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  • #2
welcome to pf!

hi tarwe! welcome to pf! :smile:
tarwe said:
but how does the =11 bit make a difference?

it doesn't :wink:

(btw, it's not 256)
 
  • #3


tiny-tim said:
hi tarwe! welcome to pf! :smile:


it doesn't :wink:

(btw, it's not 256)


thank you!
that was a typo, meant 269, was thinking in binary :/
 
  • #4
tarwe said:

Homework Statement


If [itex]\phi[/itex](x,y,z) = x3 + 2xy +yz3 find [itex]\nabla[/itex][itex]\phi[/itex] at the point P=(1,1,2) and direction of the unit normal to the surface [itex]\phi[/itex](x,y,z) = 11 at P.

Homework Equations


The Attempt at a Solution


Worked out [itex]\nabla[/itex][itex]\phi[/itex] to be 5i + 10j + 12k
Got |[itex]\nabla[/itex][itex]\phi[/itex]|= √256

so the unit n0ormal to surface is surely [itex]\frac{5+10+12}{√269}[/itex]
You mean [itex]\frac{5i+10j+12k}{√269}[/itex]

but how does the =11 bit make a difference?
It would change the exact position but not the normal to the surface.

Thanksss

Homework Statement


Homework Equations


The Attempt at a Solution

 

FAQ: Vector Calculus, Unit normal to surface help

What is vector calculus?

Vector calculus is a branch of mathematics that deals with the study of vector fields and their derivatives, such as gradient, divergence, and curl. It is commonly used in physics and engineering to model and solve problems involving quantities that have both magnitude and direction.

What is a unit normal vector to a surface?

A unit normal vector to a surface is a vector that is perpendicular to the tangent plane of the surface at a specific point. It has a magnitude of 1 and is used to determine the orientation of the surface at that point.

How is a unit normal vector calculated?

A unit normal vector can be calculated by taking the cross product of two tangent vectors to the surface at a specific point. The resulting vector will be perpendicular to both tangent vectors and will have a magnitude of 1 if the tangent vectors are also unit vectors.

What is the role of unit normal vectors in vector calculus?

Unit normal vectors are important in vector calculus as they are used to define the direction of a surface and are necessary for calculating surface integrals, which are used to find the flux of a vector field through a surface.

How can unit normal vectors be visualized?

Unit normal vectors can be visualized as arrows pointing outwards from a surface, perpendicular to the tangent plane at each point. They can also be represented as lines perpendicular to the surface in 2D cross-sections.

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