Calculus 3

Find the derivative of the vector-valued function $\mathbf{R}(t) = (f(t), g(t), h(t))$ where $f$, $g$, and $h$ are scalar functions.

Using double integrals, find the volume under a given multivariable function.

Evaluate a triple integral to find the average temperature over a defined 3D surface.

Consider the matrix A, which is $\begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}$. Find the eigenvalues and corresponding eigenvectors.

Given a function with two-dimensional input $(x, y)$ and a vector output, determine the vector at a specific input point such as $(1, 2)$ using the functions $(y^3 - 9y)$ for the x-component and $(x^3 - 9x)$ for the y-component.

Describe a curve using a position vector-valued function.

Consider the vector field $F(x,y) = -yi + xj$. To get an idea of how this vector field looks, plug in a few coordinates and plot the resultant vectors.

Consider the vector field F = ⟨(x^2)y, -x/y, xyz⟩. To find the divergence, take del dot this expression.

Consider the vector field F = ⟨(x²)y, -x/y, xyz⟩. Find the curl by taking the determinant of the matrix with i, j, k; d/dx, d/dy, d/dz; (x²)y, -x/y, xyz.

Sketch the curve whose vector equation is $r(t) = \cos(t) \mathbf{i} + \sin(t) \mathbf{j} + t \mathbf{k}$.

Using vector valued functions, describe the path of a particle or object, taking into account time as a variable.

Parametrize the same curve using different rates and understand the derivative of a position vector valued function.

Describe the path of a particle in three-dimensional space using vector valued functions.

Given the vector-valued function $r(t) = \langle 4\cos(t), 4\sin(t), 3 \rangle$, determine the curve it describes in 3D space.

Given the vector-valued function $r(t) = \langle 0, 3\cos(t), 5\sin(t) \rangle$, describe the curve in 3D space.

Given the vector-valued function $r(t) = \langle t, 3\cos(t), 5\sin(t) \rangle$, describe the curve in 3D space.

Given the vector-valued function $r(t) = \langle 6\cos(t), 0, 6\sin(t) \rangle$, describe the curve in 3D space and explain the effect of a negative $y$-component like $-t$.

Given the vector-valued function $r(t) = \langle 2\cos(t), 2\sin(t), 6e^{-t/4} \rangle$, analyze how the curve behaves in 3D space and the effect of exponential decay in the $z$-component.

Give an example of a vector-valued function r(t) and determine its domain and range in $R^3$.

Visualize a vector-valued function and analyze its behavior as it does not intersect itself within a given domain subset from $t = 5$ to $t = 20$.