Calculus 3

Using the vector cross product, determine the vector perpendicular to two given initial vectors using the right-hand rule.

Calculate the gradient vector for a given function $f(x, y)$ and describe its significance in the context of a 3D graph.

Using a topographical map, analyze the contours to plan a route through the mountains with minimal elevation changes. Discuss the importance of this analysis in winter sports like skiing or snowshoeing.

Compute the gradient of a multivariable function by finding its partial derivatives and forming a vector.

Calculate the partial derivative of a function Z with respect to X, holding Y constant.

Calculate the partial derivative of a function Z with respect to Y, holding X constant.

Explain and visualize different types of multivariable functions.

The limit of multivariable function given by the expression with three variables x, y, and z, using parametric curves for variables substitution.

Using the Extreme Value Theorem, find the global maximum and minimum values of a multivariable function $f(x, y)$ on a domain that is closed and bounded, either in the interior or along the boundary.

Find the equation of a quadric surface using the general form $Ax^2+By^2+Cz^2+Dxy+Exz+Fyz+Gx+Hy+Iz+J=0$.

Graph the hyperboloid of two sheets and explain why it does not pass through the origin, using its equation with two negative terms.

Solve problems involving different variations of axes given a three-dimensional graph and assess the suitability of the axis variation for the right-hand rule.

Determine the resulting shape when both T and S vary freely in the function representing a torus.

Make a tangent plane to a surface at a specified point by calculating the partial derivatives with respect to x and y and then substituting into the tangent plane equation, similar to the previously discussed method.

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.

Describe a curve using a position vector-valued function.

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.