What we want won't work wery well. :-( If you want to record very low frequency signals with the soundcard, you can use a similar technique. No multiplier needed, just an opamp and a 1kHz square wave generator. Chopp your signal with that squarewave with a phase inverter and record also the squarewave with the other channel. You can then retrieve your original signal and with some audio software(Audacy) you can pitch-shift to any ratio and even listen to it.
What's normally done in a dynamic signal analyzer (aka FFT spectrum analyzer) is to digitize the signal, and then multiply digitally by sine and cosine to shift the frequency. Do a search for "image rejection mixer." Of course, since you want to look at LOW frequencies, you wouldn't need to do that. If you can simply digitize the signal in its frequency range, you can use an FFT to look at the spectral components. You may wish to use windowing to keep artifacts from the abrupt edges of the sampled block from spreading throughout the FFT'd spectrum.
If you want to use a multiplier to move the spectrum up to a range where you can look at it with a spectrum analyzer that won't go to low enough frequency, you can do that, but be aware that just multiplying gives you both sum and difference frequencies. If you use a square wave, if it's a perfect square wave, you'll have images of the spectrum at the (say 1kHz) fundamental, and at all odd harmonics (3kHz, 5kHz,
7kHz, ...) If your signal of interest is narrow-band enough, that won't matter. You can insure it is narrow band enough if you low-pass-filter it before feeding it to the multiplier. You need to do that anyway, so that other frequency components don't alias into places you don't want them. If you use a square wave, you might as well use a frequency mixer instead of a multiplier. They tend to be cheaper. If you decide to filter a square wave to make it sinusoidal, use a low pass filter that has steeper cutoff than Phil's suggested 18dB/octave; that won't attenuate the 3rd harmonic much at all. It can be easier to just use a bandpass filter for a single fixed frequency. You can also just filter the OUTPUT of the mixer/multiplier. But again, if the signals of interest occupy a narrow bandwidth, it shouldn't matter. You can just let the spectrum analyzer get rid of the unwanted bands, since they are far away from the interesting stuff.
The resulting transposed frequencies will be very hard to see because they are so close together. A better solution is to digitize your low-frequency source and play it back at a faster rate thereby spreading it out across a larger bandwidth.
For example, if your frequency band of interest is 0 - 1 Hz, sample at
16 samples/s and play back at 16 k samples/s, giving 0 - 1000 Hz.
A multiplier doesn't do this as already mentioned. You'd want a single sideband modulator. This basically is a set of two multipliers operating in quadrature and with two splitter/combiners. If adjusted properly, the carrier and the other sideband are suppressed by say 30..40dB.
As mentioned, multiplication will shift the spectrum up by 1 KHZ, but not spread it out. 10 Hz input will make a line at 1010 Hz. You'll also get a reversed, mirror image spectrum that goes down from 1 KHz (another line at 990.) A square wave will work just as well, if you don't care about the additional stuff it will create around 3,5,7... KHz. Multiplication of an analog signal by a square wave (ie, by +-1) is simple and can be made very accurate.
This is useful if you have a spectrum analyzer with good resolution but that doesn't sweep all the way down to zero frequency.