Multivariable Calculus Basics Cheat Sheet

A printable reference covering multivariable functions, partial derivatives, gradients, tangent planes, double integrals, and polar coordinates for grade 12.

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Multivariable calculus studies functions that depend on two or more input variables, such as $f(x,y)$ or $f(x,y,z)$ . Students need this cheat sheet to organize the main ideas that extend single-variable calculus into higher dimensions. It is especially useful for reviewing surfaces, rates of change, optimization, and accumulation over regions. These tools appear in physics, engineering, economics, and advanced mathematics. The core ideas are partial derivatives, gradients, tangent planes, and multiple integrals. A partial derivative measures how a function changes when one variable changes and the others stay fixed. The gradient $\nabla f$ points in the direction of greatest increase and gives directional derivatives. Double integrals such as $\iint_R f(x,y)\,dA$ add values over a two-dimensional region, often using rectangular or polar coordinates.

Key Facts

For a function $z=f(x,y)$ , the partial derivative with respect to $x$ is $f_x(x,y)=\frac{\partial f}{\partial x}=\lim_{h\to 0}\frac{f(x+h,y)-f(x,y)}{h}$ .
For a function $z=f(x,y)$ , the partial derivative with respect to $y$ is $f_y(x,y)=\frac{\partial f}{\partial y}=\lim_{h\to 0}\frac{f(x,y+h)-f(x,y)}{h}$ .
The gradient of $f(x,y)$ is $\nabla f=\langle f_x,f_y\rangle$ , and for $f(x,y,z)$ it is $\nabla f=\langle f_x,f_y,f_z\rangle$ .
The directional derivative of $f$ in the unit direction $\mathbf{u}$ is $D_{\mathbf{u}}f=\nabla f\cdot \mathbf{u}$ .
The tangent plane to $z=f(x,y)$ at $(a,b,f(a,b))$ is $z=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)$ .
A double integral over a rectangular region $R=[a,b]\times[c,d]$ can be written as $\iint_R f(x,y)\,dA=\int_a^b\int_c^d f(x,y)\,dy\,dx$ .
In polar coordinates, $x=r\cos\theta$ , $y=r\sin\theta$ , and the area element becomes $dA=r\,dr\,d\theta$ .
For critical points of $f(x,y)$ , solve $f_x=0$ and $f_y=0$ , then use $D=f_{xx}f_{yy}-(f_{xy})^2$ to classify local behavior when possible.

Vocabulary

Multivariable function: A function such as $f(x,y)$ or $f(x,y,z)$ that has more than one independent input variable.
Partial derivative: A derivative like $\frac{\partial f}{\partial x}$ that measures change in one variable while holding the other variables constant.
Gradient: The vector $\nabla f$ made from the partial derivatives of $f$ , pointing in the direction of greatest increase.
Directional derivative: The rate of change of $f$ in a chosen unit vector direction $\mathbf{u}$ , given by $D_{\mathbf{u}}f=\nabla f\cdot\mathbf{u}$ .
Tangent plane: A plane that locally approximates a surface $z=f(x,y)$ near a point using $f_x$ and $f_y$ .
Double integral: An integral $\iint_R f(x,y)\,dA$ that accumulates values of a function over a two-dimensional region $R$ .

Common Mistakes to Avoid

Differentiating both variables in a partial derivative, which is wrong because $\frac{\partial f}{\partial x}$ treats $y$ as a constant and $\frac{\partial f}{\partial y}$ treats $x$ as a constant.
Using a non-unit vector in $D_{\mathbf{u}}f=\nabla f\cdot\mathbf{u}$ , which gives the wrong rate because the formula requires $\mathbf{u}$ to have length $1$ .
Forgetting the factor $r$ in polar integrals, which is wrong because the area element changes from $dA$ to $r\,dr\,d\theta$ .
Mixing the order and limits in an iterated integral, which can describe the wrong region if the bounds do not match the chosen order of integration.
Assuming $f_x=0$ and $f_y=0$ always gives a maximum or minimum, which is wrong because a critical point can also be a saddle point.

Practice Questions

1 Find $f_x$ and $f_y$ for $f(x,y)=3x^2y-4xy^3$ , then evaluate both at $(1,2)$ .
2 Find the tangent plane to $z=x^2+xy+y^2$ at the point $(1,2,7)$ .
3 Evaluate $\int_0^2\int_0^3 (x+y)\,dy\,dx$ .
4 Explain why the gradient $\nabla f(a,b)$ is perpendicular to the level curve $f(x,y)=c$ at the point $(a,b)$ .