I would like a hint on how to begin a vector calculus problem

In summary, the conversation discusses how to find the angle between the surface normal directions of two given surfaces at a specific point. The first surface is a sphere with center (0,0,0) and radius 3, and its normal vector is parallel to the radius vector from the center to the given point. The second surface can be written as the level curve of a function F(x,y,z), and the normal vector can be found by using the gradient of F and considering the direction of maximum rate of change. The second part of the question asks for the angle between the two normal vectors.
  • #1
Michael King
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0

Homework Statement


I am just confused about how the question is structured, and I am unsure on how to get the relevant information to answer the question:

Find the angle between the surface normal directions of [tex]r^{2} = 9[/tex] and [tex] x + y + z^{2} = 1 [/tex] at the joined point (2,-2,1)


Homework Equations


I am thinking both the Scalar and Vector product rules, possibly in component form.

The Attempt at a Solution



I begin by stating the definition of [tex]r^{2}[/tex]:

[tex]r^{2} = x^{2} + y^{2} + z^{2}[/tex]

I draw a sphere on paper and indicate the direction of the radius from the origin to the joined point and put the r unit vector following the direction of the radius, and I am really stuck on how to express [tex] x + y + z^{2} = 1 [/tex], and to bring the two together. It has been a LONG summer, and I was thinking that the x,y,z terms could be the components but isn't that function a scalar function? How would I get the normal direction for that? I was thinking of taking the grad of the function then taking the scalar product of the two, but I am stuck on expressing it.
 
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  • #2
So the question has two parts
- find the normal vectors to each of the surfaces given at the point p = (2,-2,1)
- what is the angle between the two vectors

For the first part:

as you hinted r^2 = 9 is an equation of a sphere with centre 0 & radius 3.

The normal to the surface, at any point p on the surface, will be parallel to the radius vector from centre 0 = (0,0,0) to p

As for the other surface you can write the surface as the level curve of F(x,y,z)
F(x,y,z) = x + y + z^2 - 1 = 0

NOw consider a tangent plane to the surface at a point p on the surface.

As F(x,y,z) is a level curve, the rate of change of F for movement in any direction in the tangent plane will be zero (why?).

Any ideas what you can use to find the direction of maximium rate of change of F, and how this relates to the last comment?
 

FAQ: I would like a hint on how to begin a vector calculus problem

What is vector calculus?

Vector calculus is a branch of mathematics that deals with the differentiation and integration of vector fields, which are functions that assign a vector to each point in a space. It is an important tool in many fields of science and engineering, including physics, engineering, and computer graphics.

How do I approach a vector calculus problem?

To begin a vector calculus problem, it is important to first understand the problem and what is being asked. Then, identify the relevant vector quantities and their relationships. Next, use the appropriate vector operations, such as dot product or cross product, to manipulate the given equations and solve for the unknown variables.

What are some common techniques used in vector calculus?

Some common techniques used in vector calculus include differentiation, integration, and the use of vector operations such as dot product, cross product, and gradient. Vector calculus also involves the use of vector fields and the concept of line and surface integrals.

How can I improve my skills in vector calculus?

To improve your skills in vector calculus, it is important to practice solving a variety of problems and familiarize yourself with different techniques and formulas. You can also seek help from a tutor or join a study group to gain a better understanding of the subject.

Are there any resources available for learning vector calculus?

Yes, there are many resources available for learning vector calculus, including textbooks, online courses, and video tutorials. You can also consult with your professor or visit your school's math or science department for additional resources and support.

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