Multivariable Calculus, Line Integral

In summary, the vector field F is given by $\mathbf{F}=\dfrac{(x, y)} {\sqrt {1-x^2-y^2}}$, and the line integral is given by $$\int_{C} \vec F \cdot dr$$. The path C is given by $$x(t)=\frac{\cos t}{1+e^t}$$ $$y(t)=\frac{\sin t}{1+e^t}$$ and the line integral is given by $$\int_0^{\infty} (F_x(x(t),y(t))x'(t)+F_y(x(t),y(
  • #1
tompenny
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Homework Statement
Calculate a line integral
Relevant Equations
$$\mathbf{F} = \dfrac{(x, y)} {\sqrt {1-x^2-y^2}}$$
The vector field F which is given by $$\mathbf{F} = \dfrac{(x, y)} {\sqrt {1-x^2-y^2}}$$

And the line integral $$ \int_{C} F \cdot dr $$C is the path of $$\dfrac{\ (\cos (t), \sin (t))}{ 1+ e^t}$$ , and $$0 ≤ t < \infty $$

How do I calculate this? Anyone got a tip/hint? many thanks
 
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  • #2
tompenny said:
Homework Statement:: Calculate a line integral
Relevant Equations:: $$\mathbf{F} = \dfrac{(x, y)} {\sqrt {1-x^2-y^2}}$$

The vector field F which is given by $$\mathbf{F} = \dfrac{(x, y)} {\sqrt {1-x^2-y^2}}$$

And the line integral $$ \int_{C} F \cdot dr $$C is the path of $$\dfrac{\ (\cos (t), \sin (t))}{ 1+ e^t}$$ , and $$0 ≤ t < \infty $$

How do I calculate this? Anyone got a tip/hint? many thanks
Remember, ##\int_C \vec F \cdot d\vec r = \int_t \vec F(\vec r(t))\cdot \vec r'(t)~dt##. So the obvious hint is plug the formulas in and see what happens. Then come back to show us where you are stuck. There is no substitute for getting your hands dirty.
 
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  • #3
My advice would be to split ##\mathbf{F}## as well the path ##C## into x and y components. For example it will be $$F_x(x,y)=\frac{x}{\sqrt{1-x^2-y^2}}$$ $$F_y(x,y)=\frac{y}{\sqrt{1-x^2-y^2}}$$while for the path C it will be $$x(t)=\frac{\cos t}{1+e^t}$$ $$y(t)=\frac{\sin t}{1+e^t}$$. Then calculate $$x'(t)$$ and $$y'(t)$$ (derivatives of x(t),y(t)with respect to t). Then combine the whole thing to $$\int_0^{\infty} (F_x(x(t),y(t))x'(t)+F_y(x(t),y(t))y'(t))dt$$.
This might look a bit scary but i think if you do all the calculations correctly and the algebra correctly some simplifications will occur (by the use of trigonometric identities).
 
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1. What is Multivariable Calculus?

Multivariable Calculus is a branch of mathematics that deals with the study of functions of multiple variables. It involves the application of calculus concepts, such as limits, derivatives, and integrals, to functions with more than one independent variable. It is used in many fields, including physics, engineering, economics, and statistics.

2. What is a Line Integral?

A Line Integral is a type of integral that is used to calculate the total value of a function along a specific curve or line. It involves breaking the curve into small segments and finding the sum of the values of the function at each point. Line integrals are commonly used in physics and engineering to calculate quantities such as work, electric flux, and magnetic flux.

3. What are the applications of Multivariable Calculus and Line Integrals?

Multivariable Calculus and Line Integrals have numerous applications in various fields. In physics, they are used to study motion, forces, and energy. In engineering, they are used to design structures and optimize systems. In economics, they are used to analyze supply and demand and optimize production. In statistics, they are used to analyze multivariate data and make predictions.

4. What is the difference between a single variable and multivariable function?

A single variable function has only one independent variable, while a multivariable function has more than one independent variable. This means that the output of a single variable function depends on only one input, while the output of a multivariable function depends on multiple inputs. Multivariable functions also have multiple partial derivatives, while single variable functions have only one derivative.

5. How is Multivariable Calculus related to other branches of mathematics?

Multivariable Calculus is closely related to other branches of mathematics, such as linear algebra and differential equations. It uses concepts from linear algebra, such as vectors and matrices, to study multivariable functions. It also provides the foundation for studying partial differential equations, which are used to model complex systems in physics and engineering.

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