When to Implicit , When Not To?

  • Thread starter magorium
  • Start date
  • Tags
    Implicit
In summary, when taking partial derivatives in multivariable calculus, you only differentiate with respect to the variable you are interested in and treat the other variables as constants. You do not use implicit differentiation.
  • #1
magorium
9
0

Homework Statement




We have a surface z^2 = 4x^2 + 2yx + 5y^2 , find the shortest distance to Origin.


Homework Equations





The Attempt at a Solution



My trouble is , i think z^2 - 4x^2 + 2yx + 4y^2 = 0 as a constraint to function L = x^2 + y^2 + z^2 (Square of distance formula. If distance is minimum , square of it should be too).

Now i will use grad(L) = Lambda*grad(surface)

But while finding the gradient of surface , i need derivatives respect to x , y and z as vector components. Now , while taking the derivative of constraint respect to x , should i use the implicit differentiaton ? My book just directly takes the differential of it respect to x , doesn't use the implicit diff.

When dealing with plane equations , multivariable calculus , when to use implicit diff and when not to ? How could i understand if z is a function of x and y or they all are three variables ?
 
Last edited:
Physics news on Phys.org
  • #2
When you take partial derivatives with respect to x, y, or z, you are, by definition, treating the other variables as constant. That is why you do not differentiate y and z, for example, with respect to x when taking the partial derivative with respect to x.
 

FAQ: When to Implicit , When Not To?

When should I use implicit methods in my research?

Implicit methods should be used when the relationship between variables is assumed to be constant or unchanging. This can be useful when studying fundamental or well-established principles. Examples include using implicit differentiation in calculus or using implicit memory in psychology experiments.

When should I avoid using implicit methods in my research?

Avoid using implicit methods when the relationship between variables is uncertain or complex. Implicit methods assume a constant relationship, so if this is not the case, it can lead to inaccurate or misleading results. Additionally, implicit methods may not be appropriate for studying novel or innovative concepts.

What are the benefits of using implicit methods in research?

One of the main benefits of using implicit methods is that they can simplify complex relationships and make them easier to understand. Implicit methods can also save time and resources by avoiding the need for explicit measurements or calculations. Additionally, implicit methods can be used to study underlying processes and mechanisms that may not be easily observed or measured explicitly.

What are the limitations of using implicit methods in research?

One limitation of using implicit methods is that they may not be appropriate for all research questions or scenarios. As mentioned, implicit methods assume a constant relationship, so if this is not the case, it can lead to inaccurate results. Additionally, implicit methods may not be suitable for studying novel or complex concepts, as they may oversimplify the relationships between variables.

How do I determine when to use implicit methods in my research?

The decision to use implicit methods should be based on the research question and the nature of the variables being studied. If the relationship between variables is well-established and assumed to be constant, implicit methods may be appropriate. However, if the relationship is uncertain or complex, it may be better to use explicit methods or a combination of both. It's important to carefully consider the limitations and benefits of implicit methods before deciding to use them in your research.

Similar threads

Lagrange multipliers understanding
Replies
8
Views
1K
Applying the implicit function theorem to a system of equations
Replies
10
Views
2K
Computing line integral using Stokes' theorem
Replies
8
Views
1K
Find the constrained maxima and minima of ##f(x,y,z)=x+y^2+2z##
Replies
2
Views
988
Determining domain for C^1 function
Replies
4
Views
803
Implicit differentiation of many variables
Replies
3
Views
2K
Optimization of the distance from the point on an ellipse
Replies
1
Views
1K
I need some help with implicit differentiaiton.
Replies
5
Views
1K
Write the tangent lines to the curve
Replies
1
Views
1K
Solve the differential equation: y′′y′+yy′+yy′′=0
Replies
5
Views
2K

Hot Threads

Recent Insights

Back
Top