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- https://math.libretexts.org/Courses/SUNY_Geneseo/Math_223_Calculus_3/01%3A_Vectors_in_SpaceA quantity that has magnitude and direction is called a vector. Vectors have many real-life applications, including situations involving force or velocity. For example, consider the forces acting on a...A quantity that has magnitude and direction is called a vector. Vectors have many real-life applications, including situations involving force or velocity. For example, consider the forces acting on a boat crossing a river. The boat’s motor generates a force in one direction, and the current of the river generates a force in another direction. Both forces are vectors. We must take both the magnitude and direction of each force into account if we want to know where the boat will go.
- https://math.libretexts.org/Courses/SUNY_Geneseo/Math_223_Calculus_3/02%3A_Vector-Valued_Functions/2.03%3A_Arc_Length_and_Curvature/2.3E%3A_Exercises_for_Section_2.316) Given \(\vecs r(t)=⟨2e^t,\, e^t \cos t,\, e^t \sin t⟩\), find the unit tangent vector \(\vecs T(t)\) evaluated at \(t=0\), \(\vecs T(0)\). 18) Given \(\vecs r(t)=⟨2e^t,\, e^t \cos t,\, e^t \sin t⟩...16) Given \(\vecs r(t)=⟨2e^t,\, e^t \cos t,\, e^t \sin t⟩\), find the unit tangent vector \(\vecs T(t)\) evaluated at \(t=0\), \(\vecs T(0)\). 18) Given \(\vecs r(t)=⟨2e^t,\, e^t \cos t,\, e^t \sin t⟩\), find the unit normal vector \(\vecs N(t)\) evaluated at \(t=0\), \(\vecs N(0)\). 29) Parameterize the curve using the arc-length parameter \(s\), at the point at which \(t=0\) for \(\vecs r(t)=e^t \sin t \,\hat{\mathbf{i}} + e^t \cos t \,\hat{\mathbf{j}}\)
- https://math.libretexts.org/Courses/SUNY_Geneseo/Math_223_Calculus_3/02%3A_Vector-Valued_Functions/2.01%3A_Vector-Valued_Functions_and_Space_CurvesOur study of vector-valued functions combines ideas from our earlier examination of single-variable calculus with our description of vectors in three dimensions from the preceding chapter. In this sec...Our study of vector-valued functions combines ideas from our earlier examination of single-variable calculus with our description of vectors in three dimensions from the preceding chapter. In this section, we extend concepts from earlier chapters and also examine new ideas concerning curves in three-dimensional space. These definitions and theorems support the presentation of material in the rest of this chapter and also in the remaining chapters of the text.
- https://math.libretexts.org/Courses/SUNY_Geneseo/Math_223_Calculus_3/04%3A_Multiple_Integration/4.01%3A_Double_Integrals_over_Rectangular_RegionsIn this section we investigate double integrals and show how we can use them to find the volume of a solid over a rectangular region in the xy-plane. Many of the properties of double integrals are sim...In this section we investigate double integrals and show how we can use them to find the volume of a solid over a rectangular region in the xy-plane. Many of the properties of double integrals are similar to those we have already discussed for single integrals.
- https://math.libretexts.org/Courses/Al_Akhawayn_University/MTH2301_Multivariable_Calculus/16%3A_Appendices/16.03%3A_Table_of_Integrals39. \(\quad \displaystyle ∫u^n\sin u\,du=−u^n\cos u+n∫u^{n−1}\cos u\,du\) 40. \(\quad \displaystyle ∫u^n\cos u\,du=u^n\sin u−n∫u^{n−1}\sin u\,du\) 70. \(\quad \displaystyle ∫\frac{\sqrt{a^2+u^2}}{u}\,...39. \(\quad \displaystyle ∫u^n\sin u\,du=−u^n\cos u+n∫u^{n−1}\cos u\,du\) 40. \(\quad \displaystyle ∫u^n\cos u\,du=u^n\sin u−n∫u^{n−1}\sin u\,du\) 70. \(\quad \displaystyle ∫\frac{\sqrt{a^2+u^2}}{u}\,du=\sqrt{a^2+u^2}−a\ln \left|\frac{a+\sqrt{a^2+u^2}}{u}\right|+C\) 71. \(\quad \displaystyle ∫\frac{\sqrt{a^2+u^2}}{u^2}\,du=−\frac{\sqrt{a^2+u^2}}{u}+\ln \left(u+\sqrt{a^2+u^2}\right)+C\)
- https://math.libretexts.org/Courses/SUNY_Geneseo/Math_223_Calculus_3/02%3A_Vector-Valued_Functions/2.02%3A_Calculus_of_Vector-Valued_FunctionsTo study the calculus of vector-valued functions, we follow a similar path to the one we took in studying real-valued functions. First, we define the derivative, then we examine applications of the de...To study the calculus of vector-valued functions, we follow a similar path to the one we took in studying real-valued functions. First, we define the derivative, then we examine applications of the derivative, then we move on to defining integrals. However, we will find some interesting new ideas along the way as a result of the vector nature of these functions and the properties of space curves.
- https://math.libretexts.org/Courses/SUNY_Geneseo/Math_223_Calculus_3/04%3A_Multiple_Integration/4.03%3A_Double_Integrals_in_Polar_CoordinatesDouble integrals are sometimes much easier to evaluate if we change rectangular coordinates to polar coordinates. However, before we describe how to make this change, we need to establish the concept ...Double integrals are sometimes much easier to evaluate if we change rectangular coordinates to polar coordinates. However, before we describe how to make this change, we need to establish the concept of a double integral in a polar rectangular region.
- https://math.libretexts.org/Courses/SUNY_Geneseo/Math_223_Calculus_3/01%3A_Vectors_in_Space/1.08%3A_Chapter_1_Review_Exercises\( x=k\) trace: \( k^2=y^2+z^2\) is a circle, \( y=k\) trace: \( x^2−z^2=k^2\) is a hyperbola (or a pair of lines if \( k=0), z=k\) trace: \( x^2−y^2=k^2\) is a hyperbola (or a pair of lines if \( k=0...\( x=k\) trace: \( k^2=y^2+z^2\) is a circle, \( y=k\) trace: \( x^2−z^2=k^2\) is a hyperbola (or a pair of lines if \( k=0), z=k\) trace: \( x^2−y^2=k^2\) is a hyperbola (or a pair of lines if \( k=0\)). 22) If the boat velocity is \( 5\) km/h due north in still water and the water has a current of \( 2\) km/h due west (see the following figure), what is the velocity of the boat relative to shore?
- https://math.libretexts.org/Courses/SUNY_Geneseo/Math_223_Calculus_3/04%3A_Multiple_Integration/4.02%3A_Double_Integrals_over_General_RegionsIn this section we consider double integrals of functions defined over a general bounded region D on the plane. Most of the previous results hold in this situation as well, but some techniques need t...In this section we consider double integrals of functions defined over a general bounded region D on the plane. Most of the previous results hold in this situation as well, but some techniques need to be extended to cover this more general case.
- https://math.libretexts.org/Courses/SUNY_Geneseo/Math_223_Calculus_3/01%3A_Vectors_in_Space/1.01%3A_Vectors_in_the_Plane/1.1E%3A_Exercises_for_Section_1.1Verify that the vectors \(\vecs a, \, \vecs b,\) and \(\vecs a+\vecs b\), and, respectively, \(\vecs a, \, \vecs b\), and \(\vecs a−\vecs b\) satisfy the triangle inequality. One end of the wire is at...Verify that the vectors \(\vecs a, \, \vecs b,\) and \(\vecs a+\vecs b\), and, respectively, \(\vecs a, \, \vecs b\), and \(\vecs a−\vecs b\) satisfy the triangle inequality. One end of the wire is attached to the top of the pole and the other end is anchored to the ground \(50\) ft from the base of the pole.
- https://math.libretexts.org/Courses/SUNY_Geneseo/Math_223_Calculus_3/05%3A_Vector_Calculus/5.07%3A_Stokes_TheoremIn this section, we study Stokes’ theorem, a higher-dimensional generalization of Green’s theorem. This theorem, like the Fundamental Theorem for Line Integrals and Green’s theorem, is a generalizatio...In this section, we study Stokes’ theorem, a higher-dimensional generalization of Green’s theorem. This theorem, like the Fundamental Theorem for Line Integrals and Green’s theorem, is a generalization of the Fundamental Theorem of Calculus to higher dimensions. Stokes’ theorem relates a vector surface integral over surface S in space to a line integral around the boundary of S.
