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CalculusCollege2 topic explainers

Multivariable Calculus course

Partial derivatives, gradients and double integrals.

Topics in this course

1.Partial Derivatives
A function of several variables, such as $f(x, y) = x^2 y + 3y$ , changes differently depending on which input you vary. The partial derivative $\frac{\partial f}{\partial x}$ measures the rate of change as $x$ varies with $y$ held fixed, and $\frac{\partial f}{\partial y}$ does the reverse. You compute them with the ordinary rules, treating the other variable as a constant.
- Hold every other variable constant and differentiate as usual.
- Mixed partials $f_{xy}$ and $f_{yx}$ are equal for smooth functions.
- The gradient is perpendicular to level curves.
Quick check: For $f = 3x^2 y$ , find $f_x$ at $(1, 4)$ .
$24$
Includes a worked example, 3 common mistakes and 3 practice questions with answers. Study partial derivatives
2.Double Integrals
A double integral $\iint_R f(x, y)\,dA$ adds up $f$ over a two-dimensional region $R$ . When $f \ge 0$ it is the volume under the surface $z = f(x, y)$ ; with $f = 1$ it gives the area of $R$ . It is computed as an iterated integral: integrate with respect to one variable, treating the other as constant, then integrate the result.
- Sketch the region and decide which variable has constant limits.
- Inner limits may be functions; outer limits must be constants.
- Polar form: $dA = r\,dr\,d\theta$ , ideal for circles.
Quick check: Evaluate $\int_0^3 \int_0^2 1\,dy\,dx$ .
$6$
Includes a worked example, 3 common mistakes and 3 practice questions with answers. Study double integrals

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