Understanding Vector Calculus: Proofs and Applications

In summary, the conversation discusses solving a problem involving parts A, B, and C. The first person is unsure of how to approach parts B and C, and is asking for clarification. They mention that they have solved part A, but are unsure of how it relates to parts B and C. The second person offers some intuition for solving part B and mentions that part C involves computing the Hessian for a function with n=3. The first person is unsure about this and asks for further explanation.
  • #1
weihead
1
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Homework Statement



http://photos-f.ak.fbcdn.net/hphotos-ak-ash1/hs425.ash1/23519_677242553394_6313872_37501934_8229842_n.jpg

I'm unsure as to how to approach part b and c... I figured out part A but I don't know how it ties into part b. Is it saying that F is a function of little f composed r(x) at x^?


Intuitively I should sub in part a into part b but I'm not sure.
Also, for part c..when it says n=3, does it mean that it is the sum of all the functions of x from 1 to 3? so the function I'm computing the Hessian for would be 1/3x^2?
 
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  • #2
For b I think you are supposed to realize that grad(g(r))=g'(r)*grad(r). I'm not really sure where they are going with c.
 

1. 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 used to describe and analyze physical phenomena involving quantities that have both magnitude and direction.

2. What is a vector field?

A vector field is a mathematical concept that assigns a vector (a magnitude and direction) to each point in a given space. It can be visualized as a collection of arrows, where the direction and length of each arrow represent the direction and magnitude of the vector at that point.

3. What is a proof in vector calculus?

A proof in vector calculus is a logical and mathematical justification for a statement or theorem in the field. It involves using mathematical principles and equations to show that a certain statement is true or false.

4. What is the role of vector calculus in physics?

Vector calculus plays a crucial role in physics as it is used to describe and analyze physical quantities that have both magnitude and direction, such as force, velocity, and acceleration. It is also used in various fields of physics, including electromagnetism, fluid mechanics, and quantum mechanics.

5. Can you give an example of a real-life application of vector calculus?

One example of a real-life application of vector calculus is in fluid mechanics, where it is used to study the flow of fluids, such as air and water. Vector calculus is also used in computer graphics to create realistic images and animations by simulating the movement of particles and fluids.

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