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- https://math.libretexts.org/Bookshelves/Calculus/CLP-4_Vector_Calculus_(Feldman_Rechnitzer_and_Yeager)/03%3A_Surface_Integrals/3.03%3A_Surface_IntegralsWe are now going to define two types of integrals over surfaces.
- https://math.libretexts.org/Courses/University_of_Maryland/MATH_241/05%3A_Vector_Calculus/5.07%3A_Surface_IntegralsIf we wish to integrate over a surface (a two-dimensional object) rather than a path (a one-dimensional object) in space, then we need a new kind of integral. We can extend the concept of a line integ...If we wish to integrate over a surface (a two-dimensional object) rather than a path (a one-dimensional object) in space, then we need a new kind of integral. We can extend the concept of a line integral to a surface integral to allow us to perform this integration. Surface integrals are important for the same reasons that line integrals are important. They have many applications to physics and engineering, and they allow us to expand the Fundamental Theorem of Calculus to higher dimensions.
- https://math.libretexts.org/Courses/Mission_College/Math_4A%3A_Multivariable_Calculus_(Kravets)/05%3A_Vector_Calculus/5.07%3A_Surface_IntegralsIf we wish to integrate over a surface (a two-dimensional object) rather than a path (a one-dimensional object) in space, then we need a new kind of integral. We can extend the concept of a line integ...If we wish to integrate over a surface (a two-dimensional object) rather than a path (a one-dimensional object) in space, then we need a new kind of integral. We can extend the concept of a line integral to a surface integral to allow us to perform this integration. Surface integrals are important for the same reasons that line integrals are important. They have many applications to physics and engineering, and they allow us to expand the Fundamental Theorem of Calculus to higher dimensions.
- https://math.libretexts.org/Under_Construction/Purgatory/MAT-004A_-_Multivariable_Calculus_(Reed)/05%3A_Vector_Calculus/5.07%3A_Surface_IntegralsIf we wish to integrate over a surface (a two-dimensional object) rather than a path (a one-dimensional object) in space, then we need a new kind of integral. We can extend the concept of a line integ...If we wish to integrate over a surface (a two-dimensional object) rather than a path (a one-dimensional object) in space, then we need a new kind of integral. We can extend the concept of a line integral to a surface integral to allow us to perform this integration. Surface integrals are important for the same reasons that line integrals are important. They have many applications to physics and engineering, and they allow us to expand the Fundamental Theorem of Calculus to higher dimensions.
- https://math.libretexts.org/Bookshelves/Calculus/Calculus_by_David_Guichard_(Improved)/16%3A_Vector_Calculus/16.07%3A_Surface_IntegralsAssuming that the quantities involved are well behaved, however, the flux of the vector field across the surface \({\bf r}(u,v)\) is \[ \iint\limits_{D} {\bf F}\cdot{\bf N}\,dS =\iint\limits_{D}{\bf F...Assuming that the quantities involved are well behaved, however, the flux of the vector field across the surface \({\bf r}(u,v)\) is \[ \iint\limits_{D} {\bf F}\cdot{\bf N}\,dS =\iint\limits_{D}{\bf F} \cdot \dfrac{{\bf r}_u \times {\bf r}_v}{|{\bf r}_u \times {\bf r}_v|} |{\bf r}_u \times {\bf r}_v|\,dA =\iint\limits_{D} {\bf F} \cdot ({\bf r}_u \times {\bf r}_v)\,dA. \nonumber\]
- https://math.libretexts.org/Courses/Mission_College/Math_4A%3A_Multivariable_Calculus_v2_(Reed)/16%3A_Vector_Calculus/16.06%3A_Surface_IntegralsIf we wish to integrate over a surface (a two-dimensional object) rather than a path (a one-dimensional object) in space, then we need a new kind of integral. We can extend the concept of a line integ...If we wish to integrate over a surface (a two-dimensional object) rather than a path (a one-dimensional object) in space, then we need a new kind of integral. We can extend the concept of a line integral to a surface integral to allow us to perform this integration. Surface integrals are important for the same reasons that line integrals are important. They have many applications to physics and engineering, and they allow us to expand the Fundamental Theorem of Calculus to higher dimensions.
- https://math.libretexts.org/Bookshelves/Calculus/Vector_Calculus_(Corral)/04%3A_Line_and_Surface_Integrals/4.04%3A_Surface_Integrals_and_the_Divergence_TheoremWe will now learn how to perform integration over a surface in \(\mathbb{R}^3\) , such as a sphere or a paraboloid. Recall from Section 1.8 how we identified points \((x, y, z)\) on a curve \(C\) in \...We will now learn how to perform integration over a surface in \(\mathbb{R}^3\) , such as a sphere or a paraboloid. Recall from Section 1.8 how we identified points \((x, y, z)\) on a curve \(C\) in \(\mathbb{R}^3\) , parametrized by \(x = x(t), y = y(t), z = z(t), a ≤ t ≤ b\), with the terminal points of the position vector.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_IV%3A_Multivariable_Calculus/03%3A_Vector_Calculus/3.06%3A_Surface_AreaIf we wish to integrate over a surface (a two-dimensional object) rather than a path (a one-dimensional object) in space, then we need a new kind of integral. We can extend the concept of a line integ...If we wish to integrate over a surface (a two-dimensional object) rather than a path (a one-dimensional object) in space, then we need a new kind of integral. We can extend the concept of a line integral to a surface integral to allow us to perform this integration. Surface integrals are important for the same reasons that line integrals are important. They have many applications to physics and engineering, and they allow us to expand the Fundamental Theorem of Calculus to higher dimensions.
- https://math.libretexts.org/Bookshelves/Calculus/Calculus_(OpenStax)/16%3A_Vector_Calculus/16.06%3A_Surface_IntegralsIf we wish to integrate over a surface (a two-dimensional object) rather than a path (a one-dimensional object) in space, then we need a new kind of integral. We can extend the concept of a line integ...If we wish to integrate over a surface (a two-dimensional object) rather than a path (a one-dimensional object) in space, then we need a new kind of integral. We can extend the concept of a line integral to a surface integral to allow us to perform this integration. Surface integrals are important for the same reasons that line integrals are important. They have many applications to physics and engineering, and they allow us to expand the Fundamental Theorem of Calculus to higher dimensions.
- https://math.libretexts.org/Courses/City_College_of_San_Francisco/CCSF_Calculus/16%3A_Vector_Calculus/16.07%3A_Surface_IntegralsIf we wish to integrate over a surface (a two-dimensional object) rather than a path (a one-dimensional object) in space, then we need a new kind of integral. We can extend the concept of a line integ...If we wish to integrate over a surface (a two-dimensional object) rather than a path (a one-dimensional object) in space, then we need a new kind of integral. We can extend the concept of a line integral to a surface integral to allow us to perform this integration. Surface integrals are important for the same reasons that line integrals are important. They have many applications to physics and engineering, and they allow us to expand the Fundamental Theorem of Calculus to higher dimensions.
- https://math.libretexts.org/Bookshelves/Calculus/Calculus_(Guichard)/16%3A_Vector_Calculus/16.07%3A_Surface_Integrals\[ \begin{align} \iint\limits_{D} {\bf F}\cdot{\bf N}\,dS &=\iint\limits_{D}{\bf F}\cdot {{\bf r}_u\times{\bf r}_v\over|{\bf r}_u\times{\bf r}_v|} |{\bf r}_u\times{\bf r}_v|\,dA \\[4pt] &=\iint\limits...\[ \begin{align} \iint\limits_{D} {\bf F}\cdot{\bf N}\,dS &=\iint\limits_{D}{\bf F}\cdot {{\bf r}_u\times{\bf r}_v\over|{\bf r}_u\times{\bf r}_v|} |{\bf r}_u\times{\bf r}_v|\,dA \\[4pt] &=\iint\limits_{D}{\bf F}\cdot ({\bf r}_u\times{\bf r}_v)\,dA. \end{align} \nonumber \]
