# Two problems involving vector calculus-

• iqjump123
In summary: Please provide a summary of the following conversation.In summary, the electrostatic potential at P arising form a dipole of unit strength at O, oriented in the direction of the unit vector n, is given by V= n\stackrel{OP\rightarrow}{}/r^3. Find the value of del^2V
iqjump123

## Homework Statement

(Sorry for the confusing font- I tried to figure out using the latex reference to indicate a vector, but couldn't do it- please view the "superscripted" variable as a vector. Thanks)

A. The electrostatic potential at P arising form a dipole of unit strength at O, oriented in the direction of the unit vector n, is given by V= n$\stackrel{OP\rightarrow}{}$/r^3, where r=OP and n coincides with the direction of the positive z axis. Find the value of

del^2V

B. Let F be the gravitational field of a mass oriented at the origin O and is expressed by,

F=c$\stackrel{OA\rightarrow}{}$/r^3
where c is a constant and r is the distance OA.
Find the value of
Curl F

## Homework Equations

del^2V=d^2v/dx^2+d^2v/dy^2+d^2v/dz^2

the curl of a function involves the determinant of the matrix [i j k; deriv_x deriv_y deriv_z; f_x f_y f_z] (wrote in MATLAB form)

## The Attempt at a Solution

Sorry about putting two problems together- I figured since it is of similar origin, and the problem itself put it as if it is two parts to a problem. If it must be separated, that can be done.

A. I have managed to get to the point where I can get the expression of V= <0,0,1>$\cdot$<x,y,z> /Magnitude of OP ^3.
Now do I just have to get the 2nd partial derivative of this function based on the three coordinate systems x y and z? I did it but looked pretty ugly (couldn't simplify), and wanted to make sure.

B. For this part, it looked like the function was:
F=c*<x,y,z>/(magnitude of OA)^3
I went through and calculated the curl of this function based on the formula above, but got 0. is this correct?

A) Potentials are not vector fields but scalar fields, are you saying that $v=r^{-3}$? you know that $r=\sqrt{x^{2}+y^{2}+z^{2}}$, so now you have a function of x,y and z.

b)not done this calculation but curl is an indication of vorticity which is now much the field rotates (sort of...), if everything is coming from the centre...

hunt mat- thanks for your input!
a) I think my attempt at latex input failed miserably..
the function of v is
v= (n_unitvector*OP_vector)/r^3, and yes, r=(√x2+y2+z2)^3, so the final expression is
z/((√x2+y2+z2)^3), where x2=x^2- getting the del^2 of this function is what I need to do right?

Basically you're on the right tranck.

## 1. What is vector calculus?

Vector calculus is a branch of mathematics that deals with the study of vectors and how they behave in three-dimensional space. It involves the use of mathematical operations such as differentiation and integration to analyze and manipulate vectors.

## 2. What are some real-world applications of vector calculus?

Vector calculus has a wide range of applications in many fields, including physics, engineering, and computer graphics. Some common applications include analyzing the motion of objects in three-dimensional space, calculating electromagnetic fields, and creating 3D models and animations.

## 3. What are the two main problems involving vector calculus?

The two main problems involving vector calculus are the gradient problem and the divergence problem. The gradient problem involves finding the direction and rate of change of a function in three-dimensional space, while the divergence problem involves calculating the flow of a vector field through a given surface.

## 4. How is vector calculus used in physics?

Vector calculus is an essential tool for solving problems in physics, particularly in mechanics and electromagnetism. It is used to describe the motion of objects in three-dimensional space, analyze forces and motion, and calculate the electric and magnetic fields generated by charged particles.

## 5. What are some common techniques used in vector calculus?

Some common techniques used in vector calculus include the use of vector fields, line integrals, surface integrals, and the gradient, divergence, and curl operators. These techniques are used to analyze and manipulate vectors in three-dimensional space and solve various problems in mathematics and physics.

• Calculus and Beyond Homework Help
Replies
9
Views
1K
• Calculus and Beyond Homework Help
Replies
4
Views
1K
• Calculus and Beyond Homework Help
Replies
8
Views
2K
• Calculus and Beyond Homework Help
Replies
6
Views
487
• Calculus and Beyond Homework Help
Replies
8
Views
1K
• Calculus and Beyond Homework Help
Replies
3
Views
1K
• Calculus and Beyond Homework Help
Replies
8
Views
1K
• Calculus and Beyond Homework Help
Replies
1
Views
1K
• Calculus and Beyond Homework Help
Replies
9
Views
1K
• Calculus and Beyond Homework Help
Replies
9
Views
955