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  • https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Active_Calculus_(Boelkins_et_al.)/09%3A_Multivariable_and_Vector_Functions/9.07%3A_Derivatives_and_Integrals_of_Vector-Valued_Functions
    A vector-valued function determines a curve in space as the collection of terminal points of the vectors r(t). If the curve is smooth, it is natural to ask whether r(t) has a derivative. In the sa...A vector-valued function determines a curve in space as the collection of terminal points of the vectors r(t). If the curve is smooth, it is natural to ask whether r(t) has a derivative. In the same way, our experiences with integrals in single-variable calculus prompt us to wonder what the integral of a vector-valued function might be and what it might tell us. We explore both of these questions in detail in this section.
  • https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Active_Calculus_(Boelkins_et_al.)/11%3A_Multiple_Integrals/11.08%3A_Triple_Integrals_in_Cylindrical_and_Spherical_Coordinates
    The spherical coordinates of a point P in 3-space are \rho (rho), \theta\text{,} and \phi (phi), where \rho is the distance from P to the origin, \theta is the angle that t...The spherical coordinates of a point P in 3-space are \rho (rho), \theta\text{,} and \phi (phi), where \rho is the distance from P to the origin, \theta is the angle that the projection of P onto the xy-plane makes with the positive x-axis, and \phi is the angle between the positive z axis and the vector from the origin to P\text{.} When P has Cartesian coordinates (x,y,z)\text{,} the spherical coordinates are given by
  • https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Active_Calculus_(Boelkins_et_al.)/09%3A_Multivariable_and_Vector_Functions/9.04%3A_The_Cross_Product
    In this section, we will meet a final algebraic operation, the cross product, which again conveys important geometric information.
  • https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Active_Calculus_(Boelkins_et_al.)/10%3A_Derivatives_of_Multivariable_Functions
    http://scholarworks.gvsu.edu/books/14/
  • https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Active_Calculus_(Boelkins_et_al.)/10%3A_Derivatives_of_Multivariable_Functions/10.03%3A_Second-Order_Partial_Derivatives
    In what follows, we begin exploring the four different second-order partial derivatives of a function of two variables and seek to understand what these various derivatives tell us about the function'...In what follows, we begin exploring the four different second-order partial derivatives of a function of two variables and seek to understand what these various derivatives tell us about the function's behavior.
  • https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Active_Calculus_(Boelkins_et_al.)/09%3A_Multivariable_and_Vector_Functions/9.08%3A_Arc_Length_and_Curvature
    Given a space curve, there are two natural geometric questions one might ask: how long is the curve and how much does it bend? In this section, we answer both questions by developing techniques for me...Given a space curve, there are two natural geometric questions one might ask: how long is the curve and how much does it bend? In this section, we answer both questions by developing techniques for measuring the length of a space curve as well as its curvature.
  • https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Active_Calculus_(Boelkins_et_al.)/09%3A_Multivariable_and_Vector_Functions/9.06%3A_Vector-Valued_Functions
    In \mathbb{R}^2\text{,} a parameterization of a curve is a pair of equations x = x(t) and y = y(t) that describes the coordinates of a point (x,y) on the curve in terms of a parameter ...In \mathbb{R}^2\text{,} a parameterization of a curve is a pair of equations x = x(t) and y = y(t) that describes the coordinates of a point (x,y) on the curve in terms of a parameter t\text{.} In \mathbb{R}^3\text{,} a parameterization of a curve is a set of three equations x = x(t)\text{,} y=y(t)\text{,} and z = z(t) that describes the coordinates of a point (x,y,z) on the curve in terms of a parameter t\text{.}
  • https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Active_Calculus_(Boelkins_et_al.)/11%3A_Multiple_Integrals/11.03%3A_Double_Integrals_over_General_Regions
    \frac{1}{A(D)} \iint_R f(x,y) \, dA\text{,} where A(D) is the area of D tells us the average value of the function f on D\text{.} If f(x, y) \geq 0 on D\text{,} we can inte...\frac{1}{A(D)} \iint_R f(x,y) \, dA\text{,} where A(D) is the area of D tells us the average value of the function f on D\text{.} If f(x, y) \geq 0 on D\text{,} we can interpret this average value of f on D as the height of the solid with base D and constant cross-sectional area D that has the same volume as the volume of the surface defined by f over D\text{.}
  • https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Active_Calculus_(Boelkins_et_al.)/10%3A_Derivatives_of_Multivariable_Functions/10.04%3A_Linearization-_Tangent_Planes_and_Differentials
    One of the central concepts in single variable calculus is that the graph of a differentiable function, when viewed on a very small scale, looks like a line. We call this line the tangent line and mea...One of the central concepts in single variable calculus is that the graph of a differentiable function, when viewed on a very small scale, looks like a line. We call this line the tangent line and measure its slope with the derivative. In this section, we will extend this concept to functions of several variables.
  • https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Active_Calculus_(Boelkins_et_al.)/09%3A_Multivariable_and_Vector_Functions/9.05%3A_Lines_and_Planes_in_Space
    Here, \overrightarrow{OP} is the fixed vector shown in blue, while the direction vector \mathbf{v} is the vector parallel to the vector shown in green (that is, the green vector represents \(t...Here, \overrightarrow{OP} is the fixed vector shown in blue, while the direction vector \mathbf{v} is the vector parallel to the vector shown in green (that is, the green vector represents t\mathbf{v}\text{,} and the line is traced out by the terminal points of the magenta vector).
  • https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Active_Calculus_(Boelkins_et_al.)/09%3A_Multivariable_and_Vector_Functions/9.03%3A_Dot_Product
    In this section, we will introduce a means of multiplying vectors.

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