- #1
physicss
- 25
- 4
- Homework Statement
- Hello, what does this expression mean?
- Relevant Equations
- (Picture)
I already solved w x x/|x|
For (w1,w2,w3) and (x1,x2,x3)
For (w1,w2,w3) and (x1,x2,x3)
Then you just have to take the partial derivative wrt ##x_i## and again wrt ##x_j##.physicss said:Homework Statement: Hello, what does this expression mean?
Relevant Equations: (Picture)
I already solved w x x/|x|
For (w1,w2,w3) and (x1,x2,x3) View attachment 327170
Thanks for the answer. Would xi and xj be x1 and x2 in this case?haruspex said:Then you just have to take the partial derivative wrt ##x_i## and again wrt ##x_j##.
No. Because the function is symmetric in the three parameters, you can replace them with ##x_i##, ##x_j##, ##x_k##, where it is understood that {i,j,k}={1,2,3}, but which is which is unspecified.physicss said:Thanks for the answer. Would xi and xj be x1 and x2 in this case?
Thank youharuspex said:No. Because the function is symmetric in the three parameters, you can replace them with ##x_i##, ##x_j##, ##x_k##, where it is understood that {i,j,k}={1,2,3}, but which is which is unspecified.
For example, suppose you had the function ##x_1x_2x_3## then its partial derivative wrt ##x_i## and ##x_j## would be ##x_k##.
Edit, you might also need to assume that i, j, k are in the same cyclic order as 1, 2, 3.
Edit 2: Just realised my posts may be off the mark. I need to solve it myself first.
Edit 3:
Rereading the question, I see it does not refer to indices 1, 2, 3. That is something you assumed. So my correct answer to your post #3 is:
Yes, they are using i, j, k as the indices, not 1, 2, 3.
A partial derivative is a mathematical concept used to describe the rate of change of a function with respect to one of its variables, while holding all other variables constant.
A partial derivative is a derivative of a multivariable function, while a regular derivative is a derivative of a single variable function. This means that a partial derivative only considers the change in one variable, while a regular derivative considers the change in all variables.
Holding other variables constant in a partial derivative means that while calculating the rate of change of a function with respect to one variable, all other variables are treated as constants and do not change.
A partial derivative is calculated by taking the derivative of a function with respect to one variable, while holding all other variables constant. This is done by treating the other variables as constants and using the rules of differentiation.
Partial derivatives are used to analyze the behavior of multivariable functions and to understand how changes in one variable affect the overall function. They are also useful in optimization problems and in various fields of science, such as physics, economics, and engineering.