Line Integral Notation wrt Scalar Value function

In summary, to find the line integral of a scalar function, f, along a straight line from (0, 0, 0) to (1, 1, 0), you must parametrize the path and write f and the differential, dl, in terms of the parameter. Then, evaluate the integral using the parametrized path and the corresponding derivatives of x, y, and z.
  • #1
elements
29
0
I'm getting a bit confused by the specific notation in the question and am unsure what exactly it is asking here/how to proceed.

Homework Statement


Given a scalar function ##f## find (a) ##∫f \vec {dl}## and (b) ##∫fdl##
along a straight line from ##(0, 0, 0)## to ##(1, 1, 0)##.

Homework Equations


##f(x,y,z) = 12xy + z##
##\vec {dl} = (\vec {dx},\vec {dy},\vec {dz})##

The Attempt at a Solution


[/B]
So what I'm mainly confused about is with part a, I can't seem to understand what it's referring to thus don't know how to go about starting the integral.

Is it implying that I need to parametricize the scalar function and then take the integral wrt to dl, treating it like a vector valued function approaching it with the dot product like this $$u = f(x,y,z) = 12xy + z , v = y , w = z $$ resulting in the vector function $$\vec {r}(u,v,w) = (12xy + z) \hat {\mathbf i} + y \hat {\mathbf j} + z \hat {\mathbf k} $$ with the parametrization of $$\vec {r}(t) = (t,t,0)$$ resulting in this integral $$\int (12t^2,t,0)⋅(1,1,0) dt$$

or
am I overthinking this and I'm simply looking find the line integral over each of the differential components like this: $$\int f \vec{dl} = \int f {\mathbf dx} \hat {\mathbf i} + \int f {\mathbf dy} \hat {\mathbf j} + \int f {\mathbf dz} \hat {\mathbf k}$$
 
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  • #2
Neither.
I can rule out your first option because f is a scalar, so the integral must be a vector.
Your second option overlooks the interaction between x and y in the computation of f.

Parameterise the path from (0,0,0) to (1,1,0) and write f and ##\vec{dl}## in terms of that parameter.
 
  • #3
So since the parametrized path is ##\vec r(t) = t \hat i + t \hat j + 0 \hat k##, is the correct path to take then to evaluate the integral like so
$$\int f(x,y,z) \vec {dl} = \int_0^1 f(x(t),y(t),z(t))(\vec {dx} \hat i + \vec {dy} \hat j +\vec {dz} \hat k)$$
$$= \int_0^1 (12t^2)x'(t)dt \hat i + \int_0^1 (12t^2)y'(t)dt \hat j + \int_0^1 (12t^2)z'(t)dt \hat k ,$$ where $$(x'(t),y'(t),z'(t)) = (1,1,0)$$ ?
 
  • #4
elements said:
So since the parametrized path is ##\vec r(t) = t \hat i + t \hat j + 0 \hat k##, is the correct path to take then to evaluate the integral like so
$$\int f(x,y,z) \vec {dl} = \int_0^1 f(x(t),y(t),z(t))(\vec {dx} \hat i + \vec {dy} \hat j +\vec {dz} \hat k)$$
$$= \int_0^1 (12t^2)x'(t)dt \hat i + \int_0^1 (12t^2)y'(t)dt \hat j + \int_0^1 (12t^2)z'(t)dt \hat k ,$$ where $$(x'(t),y'(t),z'(t)) = (1,1,0)$$ ?
Yes.
 

Related to Line Integral Notation wrt Scalar Value function

1. What is line integral notation with respect to a scalar value function?

Line integral notation with respect to a scalar value function is a mathematical notation used to calculate the integral of a scalar-valued function along a given curve. It is commonly used in vector calculus and is denoted by the symbol ∫.

2. How is line integral notation different from regular integrals?

Line integral notation is different from regular integrals in that it takes into account the path of integration. In regular integrals, the path of integration is not specified, whereas in line integrals, the path must be specified in order to calculate the integral.

3. What is the significance of the path of integration in line integral notation?

The path of integration in line integral notation is significant because it determines the direction in which the integration is performed. This is important because different paths can yield different values for the integral.

4. What are the applications of line integral notation?

Line integral notation has various applications in physics, engineering, and mathematics. It is used to calculate work done by a vector field, electric and magnetic flux, and other physical quantities. It is also used in the study of potential fields and gradient fields.

5. How is line integral notation used in real-life scenarios?

Line integral notation is used in real-life scenarios to calculate the work done by a force along a curved path. This is commonly seen in physics experiments, such as calculating the work done by a magnetic field on a moving charged particle. It is also used in engineering applications, such as calculating the strain energy in a curved beam.

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