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Properties and Applications of Line Integrals

  • Scalar functions: \(F\left( {x,y,z} \right),\) \(F\left( {x,y} \right),\) \(f\left( x \right)\)
    Scalar potential: \(u\left( {x,y,z} \right)\)
    Curves: \(C,\) \({C_1},\) \({C_2}\)
    Limits of integration: \(a,\) \(b,\) \(\alpha,\) \(\beta\)
    Parameters: \(t\), \(s\)
    Polar coordinates: \(r\), \(\theta\)
    Vector field: \(\mathbf{F}\left( {P,Q,R} \right)\)
    Position vector: \(\mathbf{r}\left( s \right)\)
    Unit vectors: \(\mathbf{i},\) \(\mathbf{j},\) \(\mathbf{k},\) \(\vec{\tau}\)
    Area of a region: \(S\)
    Length of a curve: \(L\)
    Mass of a wire: \(m\)
    Density: \(\rho\left( {x,y,z} \right),\) \(\rho\left( {x,y} \right)\)
    Coordinates of the center of mass: \(\bar x,\) \(\bar y,\) \(\bar z\)
    First moments: \({M_{xy}},\) \({M_{yz}},\) \({M_{xz}}\)
    Moments of inertia: \({I_x},\) \({I_y},\) \({I_z}\)
    Volume of a solid: \(V\)
    Work: \(W\)
    Magnetic field: \(\mathbf{B}\)
    Current: \(I\)
    Electromotive force: \(\varepsilon\)
    Magnetic flux: \(\psi\)
    1. Line integral of a scalar function
      Let a curve \(C\) be given by the vector function \(\mathbf{r} = \mathbf{r}\left( s \right)\), \(0 \le s \le S,\) and a scalar function \(F\) is defined over the curve \(C\).
      The line integral of the scalar function \(F\) over the curve \(C\) is written in the form
      \({\large\int\limits_0^S\normalsize} {F\left( {\mathbf{r}\left( s \right)} \right)ds} =\) \({\large\int\limits_C\normalsize} {F\left( {x,y,z} \right)ds} =\) \({\large\int\limits_C\normalsize} {Fds},\)
      where \(ds\) is the arc length differential.
    2. The line integral of a scalar function over a union of two curves is equal to the sum of the line integrals over each curve:
      \({\large\int\limits_{{C_1} \cup {C_2}}\normalsize} {Fds} =\) \({\large\int\limits_{{C_1}}\normalsize} {Fds} + {\large\int\limits_{{C_2}}\normalsize} {Fds} \)
    3. The line integral of a scalar function over a union of two curves
    4. If a smooth curve \(C\) is parametrized by the equation \(\mathbf{r} = \mathbf{r}\left( t \right),\) \(\alpha \le t \le \beta,\) then the line integral of a scalar function is expressed by the formula
      \({\large\int\limits_C\normalsize} {F\left( {x,y,z} \right)ds} =\) \({\large\int\limits_\alpha^\beta\normalsize} {F\left( {x\left( t \right),y\left( t \right),z\left( t \right)} \right) }\) \({ \sqrt {{{\left( {x’\left( t \right)} \right)}^2} + {{\left( {y’\left( t \right)} \right)}^2} + {{\left( {z’\left( t \right)} \right)}^2}} dt} \)
    5. If \(C\) is a smooth curve lying in the \(xy\)-plane and defined by the explicit equation \(y = f\left( x \right)\), \(a \le x \le b\), then the line integral is given by the expression
      \({\large\int\limits_C\normalsize} {F\left( {x,y} \right)ds} =\) \({\large\int\limits_a^b\normalsize} {F\left( {x,f\left( x \right)} \right) }\) \({\sqrt {1 + {{\left( {f’\left( x \right)} \right)}^2}} dx} \)
    6. Line integral of a scalar function in polar coordinates
      \({\large\int\limits_C\normalsize} {F\left( {x,y} \right)ds} =\) \({\large\int\limits_\alpha^\beta\normalsize} {F\left( {r\cos \theta ,r\sin \theta } \right) }\) \({\sqrt {{r^2} + {{\left( {{\large\frac{{dr}}{{d\theta }}}\normalsize} \right)}^2}} d\theta } ,\)
      where the curve \(C\) is defined by the polar function \(r\left( \theta \right).\)
    7. Line integral of a vector field
      Let a curve \(C\) be defined by the vector function \(\mathbf{r} = \mathbf{r}\left( s \right),\) \(0 \le s \le S.\) The vector
      \({\large\frac{{d\mathbf{r}}}{{ds}}\normalsize} = \vec{\tau} =\) \(\left( {\cos \alpha ,\cos \beta ,\cos \gamma } \right)\)
      is the unit vector of the tangent line to this curve.
    8. Line integral of a vector field
      Let also a vector field \(\mathbf{F}\left( {P,Q,R} \right)\) be defined over the curve \(C\). Then the line integral of the vector function \(\mathbf{F}\) along the curve \(C\) is expressed in the form
      \({\large\int\limits_C\normalsize} {Pdx + Qdy + Rdz} =\) \({\large\int\limits_0^S\normalsize} {\big( {P\cos \alpha + Q\cos \beta }}\) \(+\;{{ R\cos \gamma } \big)ds} \)
    9. Properties of line integrals of vector fields
      \({\large\int\limits_{ – C}\normalsize} {\left( {\mathbf{F} \cdot d\mathbf{r}} \right)} =\) \( – {\large\int\limits_C\normalsize} {\left( {\mathbf{F} \cdot d\mathbf{r}} \right)}, \)
      where \(-C\) denotes the curve with the opposite orientation.
      \({\large\int\limits_C\normalsize} {\left( {\mathbf{F} \cdot d\mathbf{r}} \right)} =\) \({\large\int\limits_{{C_1} \cup {C_2}}\normalsize} {\left( {\mathbf{F} \cdot d\mathbf{r}} \right)} =\) \({\large\int\limits_{C_1}\normalsize} {\left( {\mathbf{F} \cdot d\mathbf{r}} \right)} \) \(+\;{\large\int\limits_{C_2}\normalsize} {\left( {\mathbf{F} \cdot d\mathbf{r}} \right)} ,\)
      where \(C\) is the union of the curves \({C_1}\) and \({C_2}\).
    10. If the curve \(C\) is parametrized by \(\mathbf{r}\left( t \right) =\) \(\left( {x\left( t \right),y\left( t \right),z\left( t \right)} \right),\) \(\alpha \le t \le \beta, \) then the line integral of the vector field is written as
      \({\large\int\limits_C\normalsize} {Pdx + Qdy + Rdz} =\) \({\large\int\limits_\alpha^\beta\normalsize} {\Big[ {P\left( {x\left( t \right),y\left( t \right),z\left( t \right)} \right) {\large\frac{{dx}}{{dt}}\normalsize} }}\) \({{+\; {Q\left( {x\left( t \right),y\left( t \right),z\left( t \right)} \right) {\large\frac{{dy}}{{dt}}\normalsize}} }}\) \(+\;{{R\left( {x\left( t \right),y\left( t \right),z\left( t \right)} \right){\large\frac{{dz}}{{dt}}\normalsize}} \Big]dt} \)
    11. If the curve \(C\) lies in the \(xy\)-plane and defined by the equation \(y = f\left( x \right),\) \(a \le x \le b,\) then the line integral of the vector field is given by
      \({\large\int\limits_C\normalsize} {Pdx + Qdy} =\) \({\large\int\limits_\alpha^\beta\normalsize} {\Big[ {P\left( {x,f\left( x \right)} \right) }}\) \(+\;{{ Q\left( {x,f\left( x \right)} \right){\large\frac{{df}}{{dx}}\normalsize}} \Big] dx} \)
    12. Green’s theorem
      \({\large\iint\limits_C\normalsize} {\left( {{\large\frac{{\partial Q}}{{\partial x}}\normalsize} – {\large\frac{{\partial P}}{{\partial y}}\normalsize}} \right)dxdy} =\) \({\large\oint\limits_C\normalsize} {Pdx + Qdy}, \)
      where \(\mathbf{F} =\) \(P\left( {x,y} \right)\mathbf{i} \) \(+\; Q\left( {x,y} \right)\mathbf{j}\) is a continuous vector function with continuous first partial derivatives \(\partial P/\partial y,\) \(\partial Q/\partial x\) in a some domain \(R,\) which is bounded by a closed, piecewise smooth curve \(C.\)
    13. Area of a region \(R\) bounded by a closed curve \(C\)
      \(S = {\large\iint\limits_R\normalsize} {dxdy} =\) \({\large\frac{1}{2}\oint\limits_C\normalsize} {xdy – ydx} \)
    14. Path independence of line integrals
      The line integral of a vector function \(\mathbf{F} = P\mathbf{i} \) \(+\; Q\mathbf{j} \) \(+\; R\mathbf{k}\) is said to be path independent, if and only if \(P\), \(Q\) and \(R\) are continuous in a domain \(D\) and if there exists some scalar function \(u = u\left( {x,y,z} \right)\) (a scalar potential) such that
      \(\mathbf{F} = \text{grad }u\) or \(\partial u/\partial x = P,\) \(\partial u/\partial y = Q,\) \(\partial u/\partial z = R.\)
      Then the line integral is given by
      \({\large\int\limits_C\normalsize} {\mathbf{F}\left( \mathbf{r} \right) \cdot d\mathbf{r}} =\) \({\large\int\limits_C\normalsize} {Pdx + Qdy + Rdz} =\) \(u\left( B \right) – u\left( A \right).\)
    15. Test for a conservative field
      A vector field of the form \(\mathbf{F} = \text{grad }u\) is called a conservative field. The line integral of a vector function \(\mathbf{F} = P\mathbf{i} + Q\mathbf{j} \) \(+\; R\mathbf{k}\) is path independent if and only if
      \(\text{rot }\mathbf{F} =\) \(\left| {\begin{array}{*{20}{c}} \mathbf{i} & \mathbf{j} & \mathbf{k}\\ {\large\frac{\partial }{{\partial x}}\normalsize} & {\large\frac{\partial }{{\partial y}}\normalsize} & {\large\frac{\partial }{{\partial z}}\normalsize}\\ P & Q & R \end{array}} \right|\) \(= \mathbf{0}.\)
      If the line integral is taken in the \(xy\)-plane, then the following formula for a conservative field is valid:
      \({\large\int\limits_C\normalsize} {Pdx + Qdy} =\) \( u\left( B \right) – u\left( A \right).\)
      For the two-dimensional case, the test for a conservative field can be written in the form
      \({\large\frac{{\partial P}}{{\partial y}}\normalsize} = {\large\frac{{\partial Q}}{{\partial x}}\normalsize}.\)
    16. Length of a curve
      \(L = {\large\int\limits_C\normalsize} {ds} =\) \({\large\int\limits_\alpha^\beta\normalsize} {\left| {{\large\frac{{d\mathbf{r}}}{{dt}}\normalsize}\left( t \right)} \right|dt} =\) \({\large\int\limits_\alpha^\beta\normalsize} {\sqrt {{{\left( {\large\frac{{dx}}{{dt}}\normalsize} \right)}^2} + {{\left( {\large\frac{{dy}}{{dt}}\normalsize} \right)}^2} + {{\left( {\large\frac{{dz}}{{dt}}\normalsize} \right)}^2}} dt}\)
      where \(C\) is a piecewise smooth curve defined by the position vector \(\mathbf{r}\left( t \right),\) \(\alpha \le t \le \beta.\)
      If the curve \(C\) is two-dimensional, then
      \(L = {\large\int\limits_C\normalsize} {ds} =\) \({\large\int\limits_\alpha^\beta\normalsize} {\left| {{\large\frac{{d\mathbf{r}}}{{dt}}\normalsize}\left( t \right)} \right|dt} =\) \( {\large\int\limits_\alpha^\beta\normalsize} {\sqrt {{{\left( {\large\frac{{dx}}{{dt}}\normalsize} \right)}^2} + {{\left( {\large\frac{{dy}}{{dt}}\normalsize} \right)}^2}} dt} .\)
      If the curve \(C\) lies in the \(xy\)-plane and is described by the explicit function \(y = f\left( x \right)\), \(a \le x \le b,\) then its length is given by
      \(L = {\large\int\limits_a^b\normalsize} {\sqrt {1 + {{\left( {\large\frac{{dy}}{{dx}}\normalsize} \right)}^2}} dx}. \)
    17. Length of a curve in polar coordinates
      \(L = {\large\int\limits_\alpha^\beta\normalsize} {\sqrt {{{\left( {\large\frac{{dr}}{{d\theta }}\normalsize} \right)}^2} + {r^2}} d\theta } ,\)
      where the curve \(C\) is determined by the equation \(r = r\left( \theta \right),\) \(\alpha \le \theta \le \beta\) in polar coordinates.
    18. Mass of a wire
      \(m = {\large\int\limits_C\normalsize} {\rho \left( {x,y,z} \right)ds} ,\)
      where \({\rho \left( {x,y,z} \right)}\) is the mass per unit length of the wire.
      If the curve \(C\) is parametrized by the vector function
      \(\mathbf{r}\left( t \right) =\) \(\left( {x\left( t \right),y\left( t \right),z\left( t \right)} \right),\) \(\alpha \le t \le \beta, \)
      then its mass can be computed by the formula
      \(m =\) \({\large\int\limits_\alpha^\beta\normalsize} {\rho \left( {x\left( t \right),y\left( t \right),z\left( t \right)} \right) }\) \({ \sqrt {{{\left( {\large\frac{{dx}}{{dt}}\normalsize} \right)}^2} + {{\left( {\large\frac{{dy}}{{dt}}\normalsize} \right)}^2} + {{\left( {\large\frac{{dz}}{{dt}}\normalsize} \right)}^2}} dt} .\)
      If the curve \(C\) lies in the \(xy\)-plane, then its mass is given by
      \(m = {\large\int\limits_C\normalsize} {\rho \left( {x,y} \right)ds} \)
      or
      \(m =\) \({\large\int\limits_\alpha^\beta\normalsize} {\rho \left( {x\left( t \right),y\left( t \right)} \right) }\) \({ \sqrt {{{\left( {\large\frac{{dx}}{{dt}}\normalsize} \right)}^2} + {{\left( {\large\frac{{dy}}{{dt}}\normalsize} \right)}^2}} dt} \)
      (in parametric form)
    19. Center of mass of a wire
      \(\bar x = {\large\frac{{{M_{yz}}}}{m}\normalsize},\;\) \(\bar y = {\large\frac{{{M_{xz}}}}{m}\normalsize},\;\) \(\bar z = {\large\frac{{{M_{xy}}}}{m}\normalsize}\), where
      \({M_{yz}} = {\large\int\limits_C\normalsize} {x\rho \left( {x,y,z} \right)ds},\;\)
      \({M_{xz}} = {\large\int\limits_C\normalsize} {y\rho \left( {x,y,z} \right)ds}.\;\)
      \({M_{xy}} = {\large\int\limits_C\normalsize} {z\rho \left( {x,y,z} \right)ds} .\)
    20. Moments of inertia
      The moments of inertia of a curve about the \(x\)-axis, \(y\)-axis and \(z\)-axis are given by the formulas
      \({I_x} = {\large\int\limits_C\normalsize} {\left( {{y^2} + {z^2}} \right) }\) \({\rho \left( {x,y,z} \right)ds},\;\)
      \({I_y} = {\large\int\limits_C\normalsize} {\left( {{x^2} + {z^2}} \right) }\) \({\rho \left( {x,y,z} \right)ds},\;\)
      \({I_z} = {\large\int\limits_C\normalsize} {\left( {{x^2} + {y^2}} \right) }\) \({\rho \left( {x,y,z} \right)ds}.\)
    21. Area of a region bounded by a closed curve
      \(S = {\large\oint\limits_C\normalsize} {xdy} =\) \( – {\large\oint\limits_C\normalsize} {ydx} =\) \({\large\frac{1}{2}\oint\limits_C\normalsize} {xdy – ydx} \)
    22. Area of a region bounded by a closed curve
      If the closed curve \(C\) is given in parametric form \(\mathbf{r}\left( t \right) = \left( {x\left( t \right),y\left( t \right)} \right),\) \(\alpha \le t \le \beta ,\) then the area can be calculated by the formula
      \(S = {\large\int\limits_\alpha^\beta\normalsize} {x\left( t \right){\large\frac{{dy}}{{dt}}\normalsize} dt} =\) \( – {\large\int\limits_\alpha^\beta\normalsize} {y\left( t \right){\large\frac{{dx}}{{dt}}\normalsize} dt} =\) \({\large\frac{1}{2} \int\limits_\alpha^\beta\normalsize} {\left( {x\left( t \right){\large\frac{{dy}}{{dt}}\normalsize} – y\left( t \right){\large\frac{{dx}}{{dt}}\normalsize}} \right)dt} .\)
    23. Volume of a solid formed by rotating a closed curve about the x-axis
      \(V = – \pi {\large\oint\limits_C\normalsize} {{y^2}dx} =\) \( – 2\pi {\large\oint\limits_C\normalsize} {xydy} =\) \( – {\large\frac{\pi }{2}\oint\limits_C\normalsize} {2xydy + {y^2}dx} \)
    24. Volume of a solid formed by rotating a closed curve about the x-axis
    25. Work of a field of forces
      Work done by a force \(\mathbf{F}\) on an object moving along a curve \(C\) is described by the line integral
      \(W = {\large\int\limits_C\normalsize} {\mathbf{F} \cdot d\mathbf{r}} ,\)
      where \(d\mathbf{r}\) is the unit tangent vector.
    26. Work of a field of forces
      If the object is moved along a curve \(C\) lying in the \(xy\)-plane, then the work is determined by the following formula
      \(W = {\large\int\limits_C\normalsize} {\mathbf{F} \cdot d\mathbf{r}} =\) \({\large\int\limits_C\normalsize} {Pdx + Qdy} .\)
      If the path \(C\) is specified by a parameter \(t\) (\(t\) often means time), then the formula for calculating work becomes
      \(W = {\large\int\limits_\alpha^\beta\normalsize} {\Big[ {P\left( {x\left( t \right),y\left( t \right),z\left( t \right)} \right){\large\frac{{dx}}{{dt}}\normalsize} }}\) \(+\;{{ Q\left( {x\left( t \right),y\left( t \right),z\left( t \right)} \right){\large\frac{{dy}}{{dt}}\normalsize} }}\) \(+\;{{ R\left( {x\left( t \right),y\left( t \right),z\left( t \right)} \right){\large\frac{{dz}}{{dt}}\normalsize}} \Big]dt},\)
      where \(t\) goes from \(\alpha\) to \(\beta.\)
      If the vector field \(\mathbf{F}\) is conservative and \(u\left( {x,y,z} \right)\) is the scalar potential of this field, then the work on an object moving from \(A\) to \(B\) can be found by the formula
      \(W = u\left( B \right) – u\left( A \right)\).
    27. Ampere’s law
      \({\large\oint\limits_C\normalsize} {\mathbf{B} \cdot d\mathbf{r}} = {\mu _0}I\)
      The line integral of a magnetic field \(\mathbf{B}\) around a closed path \(C\) is equal to the current \(I\) (times the coefficient \({\mu _0}\)) flowing through the area bounded by the contour \(C.\)
    28. Ampere’s law
    29. Faraday’s law
      \(\varepsilon = {\large\oint\limits_C\normalsize} {\mathbf{E} \cdot d\mathbf{r}} =\) \( – {\large\frac{{d\psi }}{{dt}}\normalsize}\)
      The electromotive force \(\varepsilon\) induced around a closed loop \(C\) is equal to the rate of change of the magnetic flux \(\psi\) passing through the loop.
    30. Faraday’s law