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- https://math.libretexts.org/Bookshelves/Calculus/Calculus_3e_(Apex)/14%3A_Appendix/14.06%3A_Section_6-
- https://math.libretexts.org/Bookshelves/Calculus/Calculus_3e_(Apex)/14%3A_AppendixContributors and Attributions Gregory Hartman (Virginia Military Institute). Contributions were made by Troy Siemers and Dimplekumar Chalishajar of VMI and Brian Heinold of Mount Saint Mary's Universi...Contributors and Attributions Gregory Hartman (Virginia Military Institute). Contributions were made by Troy Siemers and Dimplekumar Chalishajar of VMI and Brian Heinold of Mount Saint Mary's University. This content is copyrighted by a Creative Commons Attribution - Noncommercial (BY-NC) License. http://www.apexcalculus.com/
- https://math.libretexts.org/Bookshelves/Calculus/Calculus_3e_(Apex)/08%3A_Sequences_and_Series/8.04%3A_Ratio_and_Root_TestsThe comparison tests of the previous section determine convergence by comparing terms of a series to terms of another series whose convergence is known. This section introduces the Ratio and Root Test...The comparison tests of the previous section determine convergence by comparing terms of a series to terms of another series whose convergence is known. This section introduces the Ratio and Root Tests, which determine convergence by analyzing the terms of a series to see if they approach 0 "fast enough.''
- https://math.libretexts.org/Bookshelves/Calculus/Calculus_3e_(Apex)/11%3A_Vector-Valued_Functions/11.01%3A_VectorValued_FunctionsWe are very familiar with real valued functions, that is, functions whose output is a real number. This section introduces vector–valued functions – functions whose output is a vector.
- https://math.libretexts.org/Bookshelves/Calculus/Calculus_3e_(Apex)/06%3A_Techniques_of_Integration/6.02%3A_Integration_by_PartsIntegration by parts is a process that finds the integral of a product of functions in terms of the integral of their derivative and antiderivative. It is frequently used to transform the antiderivati...Integration by parts is a process that finds the integral of a product of functions in terms of the integral of their derivative and antiderivative. It is frequently used to transform the antiderivative of a product of functions into an antiderivative for which a solution can be more easily found. The rule can be derived in one line simply by integrating the product rule of differentiation.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Fundamentals_of_Matrix_Algebra_(Hartman)/01%3A_Systems_of_Linear_Equations/1.04%3A_Existence_and_Uniqueness_of_SolutionsIn the “or not” case, the constants determine whether or not infinite solutions or no solution exists. (So if a given linear system has exactly one solution, it will always have exactly one solution e...In the “or not” case, the constants determine whether or not infinite solutions or no solution exists. (So if a given linear system has exactly one solution, it will always have exactly one solution even if the constants are changed.) Let’s look at an example to get an idea of how the values of constants and coefficients work together to determine the solution type.
- https://math.libretexts.org/Bookshelves/Calculus/Calculus_3e_(Apex)/13%3A_Multiple_Integration/13.01%3A_Iterated_Integrals_and_AreaPreviously, we found that it was useful to differentiate functions of several variables with respect to one variable, while treating all the other variables as constants or coefficients. We can integr...Previously, we found that it was useful to differentiate functions of several variables with respect to one variable, while treating all the other variables as constants or coefficients. We can integrate functions of several variables in a similar way.
- https://math.libretexts.org/Courses/Monroe_Community_College/MTH_212_Calculus_III/Chapter_14%3A_Multiple_Integration/14.3%3A_Double_Integration_with_Polar_CoordinatesWe have used iterated integrals to find areas of plane regions and volumes under surfaces. Just as a single integral can be used to compute much more than "area under the curve,'' iterated integrals c...We have used iterated integrals to find areas of plane regions and volumes under surfaces. Just as a single integral can be used to compute much more than "area under the curve,'' iterated integrals can be used to compute much more than we have thus far seen. The next two sections show two, among many, applications of iterated integrals.
- https://math.libretexts.org/Bookshelves/Calculus/Calculus_3e_(Apex)/10%3A_Vectors/10.03%3A_The_Dot_ProductThe previous section introduced vectors and described how to add them together and how to multiply them by scalars. This section introduces a multiplication on vectors called the dot product.
- https://math.libretexts.org/Courses/Coastline_College/Math_C280%3A_Calculus_III_(Everett)/04%3A_Multiple_Integration/4.04%3A_Double_Integration_with_Polar_CoordinatesWe have used iterated integrals to find areas of plane regions and volumes under surfaces. Just as a single integral can be used to compute much more than "area under the curve,'' iterated integrals c...We have used iterated integrals to find areas of plane regions and volumes under surfaces. Just as a single integral can be used to compute much more than "area under the curve,'' iterated integrals can be used to compute much more than we have thus far seen. The next two sections show two, among many, applications of iterated integrals.
- https://math.libretexts.org/Bookshelves/Calculus/Calculus_3e_(Apex)/12%3A_Functions_of_Several_VariablesThis chapter studies multivariable functions, that is, functions with more than one input.
