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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_FunctionsA 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_CoordinatesThe 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_ProductIn 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/10.03%3A_Second-Order_Partial_DerivativesIn 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.)/10%3A_Derivatives_of_Multivariable_Functionshttp://scholarworks.gvsu.edu/books/14/
- 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_CurvatureGiven 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_FunctionsIn \(\mathbb{R}^2\text{,}\) 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_DifferentialsOne 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_SpaceHere, \(\overrightarrow{OP}\) 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_ProductIn this section, we will introduce a means of multiplying vectors.
