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- https://math.libretexts.org/Courses/Oxnard_College/Multivariable_Calculus
- https://math.libretexts.org/Courses/Oxnard_College/Multivariable_Calculus/03%3A_Multiple_Integration/3.05%3A_Triple_Integrals\[\begin{align*} \int_{x=0}^{x=1} \int_{y=0}^{y=x^2} \int_{z=0}^{z=y^2} xyz \, dz \, dy \, dx \\ = \int_{x=0}^{x=1} \int_{y=0}^{y=x^2} \left. \left[xy \dfrac{z^2}{2} \right|_{z=0}^{z=y^2} \right] \, d...\[\begin{align*} \int_{x=0}^{x=1} \int_{y=0}^{y=x^2} \int_{z=0}^{z=y^2} xyz \, dz \, dy \, dx \\ = \int_{x=0}^{x=1} \int_{y=0}^{y=x^2} \left. \left[xy \dfrac{z^2}{2} \right|_{z=0}^{z=y^2} \right] \, dy \, dx = \int_{x=0}^{x=1} \int_{y=0}^{y=x^2} \left( x \dfrac{y^5}{2}\right) dy \, dx = \int_{x=0}^{x=1} \left. \left[ x\dfrac{y^6}{12} \right|_{y=0}^{y=x^2}\right] dx = \int_{x=0}^{x=1} \dfrac{x^{13}}{12} dx = \left. \dfrac{x^{14}}{168}\right|_{x=0}^{x=1} = \dfrac{1}{168}, \end{align*}\]
- https://math.libretexts.org/Courses/Oxnard_College/Multivariable_Calculus/03%3A_Multiple_Integration/3.04%3A_Double_Integration_with_Polar_Coordinates/3.4E%3A_Double_Integrals_in_Polar_Coordinates_(Exercises)50) The surface of a right circular cone with height \(h\) and base radius \(a\) can be described by the equation \(f(x,y)=h-h\sqrt{\frac{x^2}{a^2}+\frac{y^2}{a^2}}\), where the tip of the cone lies a...50) The surface of a right circular cone with height \(h\) and base radius \(a\) can be described by the equation \(f(x,y)=h-h\sqrt{\frac{x^2}{a^2}+\frac{y^2}{a^2}}\), where the tip of the cone lies at \((0,0,h)\) and the circular base lies in the \(xy\)-plane, centered at the origin.
- https://math.libretexts.org/Courses/Oxnard_College/Multivariable_Calculus/02%3A_Functions_of_Multiple_Variables_and_Partial_Derivatives
- https://math.libretexts.org/Courses/Oxnard_College/Multivariable_Calculus/05%3A_Vector_Fields_Line_Integrals_and_Vector_Theorems/5.03%3A_Line_IntegralsLine integrals have many applications to engineering and physics. They also allow us to make several useful generalizations of the Fundamental Theorem of Calculus. And, they are closely connected to t...Line integrals have many applications to engineering and physics. They also allow us to make several useful generalizations of the Fundamental Theorem of Calculus. And, they are closely connected to the properties of vector fields, as we shall see.Line integrals have many applications to engineering and physics. They also allow us to make several useful generalizations of the Fundamental Theorem of Calculus. And, they are closely connected to the properties of vector fields, as we shall see.
- https://math.libretexts.org/Courses/Oxnard_College/Multivariable_Calculus/05%3A_Vector_Fields_Line_Integrals_and_Vector_Theorems/5.06%3A_Divergence_and_Curl/5.6E%3A_Divergence_and_Curl_(Exercises)These are homework exercises to accompany Chapter 16 of OpenStax's "Calculus" Textmap.
- https://math.libretexts.org/Courses/Oxnard_College/Multivariable_Calculus/04%3A_Vector-valued_Functions/4.E%3A_Chapter_4_Review_ExercisesIn each let \(f(t)=t^3,\vec{r}(t) = \langle t^2, t-1, 1 \rangle \) and \(\vec{s}(t)=\langle \sin t, e^t, t \rangle \). Find \(\vec{r}(t)\), given that \(\vec{r}''(t)=\langle \cos t, \sin t, e^t \rangl...In each let \(f(t)=t^3,\vec{r}(t) = \langle t^2, t-1, 1 \rangle \) and \(\vec{s}(t)=\langle \sin t, e^t, t \rangle \). Find \(\vec{r}(t)\), given that \(\vec{r}''(t)=\langle \cos t, \sin t, e^t \rangle\), \(\vec{r}'(0) =\langle 0,0,0 \rangle \text{ and }\vec{r}(0)=\langle 0,0,0 \rangle\).
- https://math.libretexts.org/Courses/Oxnard_College/Multivariable_Calculus/01%3A_Vectors_and_the_Geometry_of_Space/1.03%3A_The_Dot_ProductThe dot product essentially tells us how much of the force vector is applied in the direction of the motion vector. The dot product can also help us measure the angle formed by a pair of vectors and t...The dot product essentially tells us how much of the force vector is applied in the direction of the motion vector. The dot product can also help us measure the angle formed by a pair of vectors and the position of a vector relative to the coordinate axes. It even provides a simple test to determine whether two vectors meet at a right angle.
- https://math.libretexts.org/Courses/Oxnard_College/Multivariable_Calculus/04%3A_Vector-valued_Functions/4.05%3A_Arc_Length_and_CurvatureIn this section, we study formulas related to curves in both two and three dimensions, and see how they are related to various properties of the same curve. For example, suppose a vector-valued functi...In this section, we study formulas related to curves in both two and three dimensions, and see how they are related to various properties of the same curve. For example, suppose a vector-valued function describes the motion of a particle in space. We would like to determine how far the particle has traveled over a given time interval, which can be described by the arc length of the path it follows.
- https://math.libretexts.org/Courses/Oxnard_College/Multivariable_Calculus/02%3A_Functions_of_Multiple_Variables_and_Partial_Derivatives/2.01%3A_Introduction_to_Functions_of_Multiple_VariablesThis function might represent the temperature over a given time interval, the position of a car as a function of time, or the altitude of a jet plane as it travels from New York to San Francisco. For ...This function might represent the temperature over a given time interval, the position of a car as a function of time, or the altitude of a jet plane as it travels from New York to San Francisco. For example, temperature can depend on location and the time of day, or a company’s profit model might depend on the number of units sold and the amount of money spent on advertising.
- https://math.libretexts.org/Courses/Oxnard_College/Multivariable_Calculus/04%3A_Vector-valued_Functions/4.04%3A_Motion_in_Space/4.4E%3A_Exercises_for_Section_12.3As the circle rolls, it generates the cycloid \(\vecs r(t)=(ωt−\sin(ωt))\,\hat{\mathbf{i}}+(1−\cos(ωt))\,\hat{\mathbf{j}}\), where \(\omega\) is the angular velocity of the circle and \(b\) is the rad...As the circle rolls, it generates the cycloid \(\vecs r(t)=(ωt−\sin(ωt))\,\hat{\mathbf{i}}+(1−\cos(ωt))\,\hat{\mathbf{j}}\), where \(\omega\) is the angular velocity of the circle and \(b\) is the radius of the circle: The velocity at \(t=1\) sec is \(\vecs v(1)=5\,\hat{\mathbf{j}}\) and the position of the object at \(t=1\) sec is \(\vecs r(1)=0\,\hat{\mathbf{i}}+0\,\hat{\mathbf{j}}+0\,\hat{\mathbf{k}}\).
